THE PHASE RULE 


A.C.D.RIVETT 


} 














THE PHASE RULE 


AND THE STUDY OF 


HETEROGENEOUS EQUILIBRIA 


AN INTRODUCTORY STUDY 
BY 
Pee 1.. RIVET T 


M.A., B.Sc. (Oxon.), D.Sc. (MEzB.) 


ASSOCIATE PROFESSOR OF CHEMISTRY IN THE UNIVERSITY OF MELBOURNE 


OXFORD 
AT THE CLARENDON PRESS 
1923 


OXFORD UNIVERSITY PRESS 
London Edinburgh Glasgow Copenhagen 
New York Toronto Melbourne Cape Town 
Bombay Calcutta Madras Shanghai 
HUMPHREY MILFORD 
Publisher to the University 


2568 de 


Printed in England 


CONTENTS 


CHAPTER 


Preface 


The Phase Rule . 


. The Principle of Le Chatelier and Braun 
. One-Component (or Unary) Systems 


. Two-Component (or Binary) Systems 


Three-Component (or Ternary) Systems 


. Four-Component (or Quaternary) Systems 


. Some Thermodynamical Considerations 


VIII. 


A Discussion on Binary and Ternary Systems 
of Mixed Crystals with Illustrations of the 
Graphical Use of the ¢-function 


187 











PREFACE 


THERE are many reasons why the study of heterogeneous 
equilibria has received insufficient attention in English- 
speaking countries. Of these, perhaps the chief is the 
unattractive nature of the experimental procedure involved 
in ‘solubility work’. There is required so often a long 
series of quantitative determinations carried out strictly in 
accordance with a specified method. The discovery and 
perfecting of the method may have been of high interest : 
its application time after time is mere routine, and routine 
seldom makes appeal to an original mind. Few scientific 
investigators, particularly of the University class, can 
employ analysts who might be content to accept labour 
of this kind. 

Then again the study of heterogeneous equilibria is one 
of those rather rare cases in which theory has outstripped 
experiment, and in which, moreover, theory has been built 
upon so firm a foundation that experimental confirmation 
or illustration often enough seems unnecessary. When 
there is so much of high significance in other sections of 
chemical science demanding experimental investigation as 
a preliminary to any theorizing, it is not surprising that 
heterogeneous equilibria find so few practical students. 

There are circumstances, however, in which a certain 

path of laboratory investigation may rightly demand close 


6 Preface 


attention from chemists for quite other reasons than its 
intrinsic interest, and there can be no doubt that with 
respect to the study of heterogeneous systems such cir- 
cumstances exist to-day. The immense waste of war must 
be made good as rapidly as possible, and as many methods 
of production of wealth must be employed as_ possible. 
The primary industries doubtless rank first in importance, 
but very near to them come the distinctively chemical 
manufactures. Of comparatively recent development even 
In the most advanced countries, and of only partial develop- 
ment in any country, these are destined to play a huge 
part, not only in making up for the losses resulting from 
the Great War, but also in determining the material 
positions of nations in the reconstructed world. 

Now scarcely a single chemical manufacture deals solely 
with homogeneous systems. In the production of prac-— 
tically every chemical substance, organic or inorganic, 
several phases enter, usually liquid or solid, but often 
enough vaporous. It is a truism that the most efficient 
handling of them can result only from systematic study : 
yet it is a fact that such study has been given only in 
isolated cases to the heterogeneous equilibria of works 
practice. To the time factor in reactions attention is 
inevitably directed: but on the equally important side of 
equilibrium but little scientific study is expended. That 
for manufacturing a given article @ process may be dis- 
covered without full laboratory investigation of the rela- 
tions between the components and phases concerned has 
been often enough shown: but that ¢he one and only most 


° Preface 7 


efficient way of getting at the desired result will be dis- 
covered by ‘hit and miss’ methods is a mere matter of 
chance, and the odds are enormously against success. At 
the best success will always be delayed. 

The study of heterogeneous equilibria, from the stand- 
points of the Phase Rule and the Principle of Le Chatelier 
and Braun, is one which every manufacturer would do 
well to require from his chemical staff. Whether or not 
costly evaporations are necessary, or recrystallizations, or 
extractions with solvents, only systematic work will show. 
The cost of such investigation will usually be slight com- 
pared with the gains that will be made possible. 

The following pages give nothing but an introduction 
to the subject. They do not constitute either a treatise 
or a book of reference. The aim has been to discuss types 
of systems which may be met and ways in which such 
systems may be graphically represented, and to give some 
examples of the manner in which conclusions of practical 
importance may be deduced. _ | 

A difficulty immediately arises in presenting the subject 
in such a way that conclusions shall conform throughout 
strictly to the basic principles summarized in the Phase 
Rule. This difficulty must have confronted all teachers 
and will be obvious to the student after he has read a little 
of the subject. It is best illustrated by a specific simple 
example such as, say, that represented in the typical 
diagram, wlfich may be found in any text-book at all on 
the Phase Rule, of a certain portion of a binary system, 
namely, that portion where temperature and phase com- , 


8 Preface ° 


positions vary while pressure is maintained at such a value 
that vapour cannot form. The lines in the accompanying 
figure repeat the familiar representation, the details of 
which need not be discussed here. 

The areas 4’DC and B’HC are two-phase, univariant 
(condensed) systems. This means that if one arbitrarily 
assigns a value (within certain limits) to any one of the 
three variables, temperature, composition of liquid or com- 
position of solid, the other 
two are single-valued con- 
stants of nature, and may 
not be varied consistently 
with the continued existence 





of the two-phase system. 
Hence, with variation in 








composition of liquids (solu- 
tions) along A/C or BC 
| there must be regular 





Temperature 











Composition variation in composition of 

solids in equilibrium, and 

vice versa. This means, however, that the relations 
represented in the figure are impossible, since the figure 
implies that solid pure B, or A, may be in equilibrium 
not only with liquid of composition B’, or A’, but with 
any liquid at all along B’C or A’C, respectively, as 
indicated by the tie-lines, In other words, arbitrary 
selection of the composition of the solid phase has not 
settled definitely and without ambiguity the temperature 
and composition of liquid at which alone the two phases 


Preface 9 


% 
may continue to coexist. Strictly speaking, therefore, the 
lines giving compositions and temperatures of solid phases 
cannot be vertical. They must incline, however slightly, 
in the ways shown by the interrupted lines in the figure. 
This comes to the same thing as saying that there cannot 
be equilibrium between two phases unless each compo- 
nent is present in each phase, and in that form it is more 
or less axiomatic. A precisely similar argument may be 
applied to the region of two solid phases below DCE. 

The fact that in an enormous number of cases the 
inclination of 4’D, or B’E, is too small to be detected 
experimentally is usually taken as justification for neglect- 
ing this mere ‘theoretical fact’, and as a rule the case 
shown in the figure is considered to be quite distinct from 
one in which the slope of 4d’) or B’FH is so great that 
experiment demonstrates it, or as it is more usually put, 
distinct from one where mixed crystals, or solid solutions, 
are formed. Actually there is no difference, except in 
degree. The case of solid solutions is the universal case, 
not only in binary, but also in higher, systems; the only 
one to be considered, strictly speaking, when principles are 
under discussion, as in a student's text-book.* 

Nevertheless, one loses much from the didactic stand- 
point in entering upon the discussion of solid solutions at 
once. It is better to ignore at first the complications 
which they introduce. The customary method of treatment 
is the best, and it is followed here ; but this can be justified 
only if the admission be definitely made that in such 

* Cf. Chemical News, 1921, 123, 251. 


10 Preface 


treatment any mevitable slight variation in the composition of 
the solid phase is deliberately left out of consideration. Those 
cases where this decision involves more than a ‘negligible ; 
error from the experimental standpoint are considered in 
a cursory introductory manner in Chapter VIII, where 
further reference is made to the general question: 

The student who is interested in heterogeneous equilibria 
should not fail to become familiar with the various develop- 
ments worked out by Roozeboom, Schreinemakers, van 
‘t Hoff, Meyerhoffer, Bancroft, Janecke and many others, 
and described in numerous memoirs in chemical literature, 
particularly in the Zeitschrift fiir physikalische Chemie. The 
thermodynamic basis, primarily due to Willard Gibbs 
(Scientific Papers, i, pp. 62-100), will also amply repay study. 

Of literature other than original ‘memoirs in chemical 
periodicals, the most important is the standard work begun 
by Roozeboom and continued by Schreinemakers and 
others, though not yet complete, Die heterogenen Gleich- 
gewichte vom Standpunkte der Phasenlehre (Vieweg und Sohn, 
Braunschweig). ‘To this the author is very deeply in- 
debted. He has made use of several of the typical diagrams 
contained in it. ss 

Janecke’s Gesdttigle Salzlisungen (Knapp, Halle a. 8.) 
gives a concise account of the chief characteristics in an 
important though limited section; but certainly the best 
and most detailed treatment of the kind is to be found in 
Clibbens’s Principles of the Phase Theory (Macmillan & Co.). 
Though this is too restricted in scope (dealing only with 
condensed systems) to be regarded as a suitable general 


Preface abit 


introduction for a student, it should be read by any one 
wishing to specialize in ‘salt chemistry’. Findlay’s Phase 
Rule and its Applications (Longmans, Green & Co.) covers 
considerable ground, especially in one- and two-component 
systems, though the omission of references to typical space 
diagrams showing the relations between all the variables 
in these respective cases makes it difficult for a student 
coming to the subject for the first time to obtain a clear 
idea of the correlation between all the projections and 
sections that are considered. Bancroft’s Phase Lule is one 
of the earlier standard books, no longer readily obtainable, 
and a great deal that is fundamental will also be found in 
Ostwald’s Lehrbuch der allgemeinen Chemie. 

It is hoped that these chapters, based upon lectures 
delivered in the University of Melbourne in 1920, will aid 
in attracting students to an insufficiently cultivated field, 
especially those students whose later years may be devoted 
to the initiation and development of chemical manufacturing 
processes. 

While taking sole responsibility for any errors that may 
be found in the book, the author has great pleasure in 
acknowledging his indebtedness to Professor Orme Masson, 
C.B.E., M.A., D.Sc., F.R.S., for the assistance and advice 
that have always been available to him; to Mr. E. F. J. 
Love, M.A., D.Sc, F.R.A.S., for very kindly reviewing the 
chapter dealing with certain matters of elementary thermo- 
dynamics; and to Mr. N. V. Sidgwick, M.A., Se.D., F.RB.S., 
for his interest while the book was in the press. 


MELBOURNE, A.C. D. RIVETT. 
July, 1921. 





CHAPTER I 


THE PHASE RULE 


The term ‘phase’. A portion of matter, homogeneous in 
the sense that its smallest mechanically isolable parts are 
indistinguishable from one another physically or chemically, 
was called by Willard Gibbs a ‘ phase’. 

‘We may call such bodies as differ in composition or 
state different phases of the matter considered, reearding all 
bodies which differ only in quantity and form as different 
examples of the same phase’ (Sczentific Papers, 1. 96). 

Homogeneity in the sense intended is a term applicable 
to a mixture of gases or to a solution of one substance in 
another. Mechanical methods, such as are effective in 
separating relatively large dissimilar masses, will be far too 
coarse to discriminate between molecules. A statement of 
homogeneity in such mixtures is parallel with a statement 
of definite temperature: in the latter case a thermometer is 
too big an instrument to show differences in temperature 
of individual molecules, though these undoubtedly exist. 

Any phase will then be separated in space (the boun- 
daries being quite definite) from every other homogeneous 
but different portion of matter. In the simplest case of 
all, that of a single chemical substance, it is clear in the 
first place that possible phases may be solid, liquid or 
gaseous. For every pure substance there must be at least 
these three phases capable of existence, though in some 
cases 16 may be beyond our means actually to realize the 
conditions of their existence. Since, however, numerous 
substances, such as sulphur, phosphorus and tin, may exist, 


14 The Phase Rule 


according to conditions, in different crystalline forms, that 
‘is, as distinct solids, it follows that for any single pure 
chemical species, three represents the minimum number of 
phases that may be expected, while the maximum number 
cannot be foreseen. 

Specification of phases. To describe completely a single 
phase, whether solid, liquid or gaseous, of some one 
chemical substance, assuming effects (if any) of gravita- 
tional, electrical, magnetic and capillary (or surface) forces 
to be of a negligible order of magnitude, it is necessary and 
sufficient to state the volume occupied: by unit mass, the 
pressure and the temperature.* Knowing these, it is 
possible always to reconstitute the system. 

All our experience points to the conclusion that there 
will exist a relation between these quantities which may be 
expressed in the general form : : 


Kes Pf a 

although it may be beyond us in the present state of knoyw- 
ledge to give a precise statement of the relation in any par- 
ticular case. Granting, however, that there will always be 
such a relation between volume, pressure and temperature, it 
is obvious that a knowledge of any two of these will suffice 
to give full information regarding the system, since the 
third will depend upon these two and be expressible in 
terms of them. 

The term ‘variable’. Since a particular phase may be 
able to exist under various conditions of pressure, tempera- 
ture and specific volume, it is usual and convenient to- 


* Correctly speaking, the general relation between the volume, 
entropy and energy is required; for from it the relation between 
volume, temperature and pressure may be deduced, whereas the con- 
verse is not true. (See Scientific Papers of J. Willard Gibbs, Biog. Note, 
xv and xvi.) The more familiar course has been followed in the text, 


The Phase Rule 15 


refer to these factors, collectively, as- the ‘variables’ of 
the phase. 

Single phases of one component, Attention will first be 
confined to phases of one and the same chemical substance, 
and this substance will be termed the ‘component’. 
A fuller discussion of the term will be given presently. 

As just stated, the condition of a single phase is com- 
pletely determined when any two of the three possible 
variables are specified, for the third is immediately settled 
in terms of these two by a relation fv, p, T) = O. If, 
however, only one of them be specified, an infinite number 
of pairs of values may be found for the other two which 
will satisfy the necessary relation. Hence in fixing the 
specification of the system one is free to select arbitrarily 
a value for one or other of the remaining two variables ; 
this being done, the value of the third is also settled. 

The term ‘degree of freedom’. It is usual to say that 
such a system has one ‘degree of freedom’. Actually the 
freedom is not, as implied, infinite: 1¢ exists only over a 
specific range, often very wide, but nevertheless limited. 
Thus, for a vapour under a given pressure, one’s freedom 
in fixing upon a volume is curtailed by the fact that for 
certain volumes (and consequent temperatures) the sub- 
stance under consideration may condense partially to a 
liquid. In fact, the ‘ freedom ’ of selecting at will a second 
variable is confined to those values of it under which the 
given chemical substance may continue to exist as, and 
only as, the single phase. The region of selection may be 
limited as regards both high and low values of the chosen 
variable ; or it may be limited in one direction only. The 
determination of the limits of such selections, or boundaries 
of existence of specified phases, will occupy much attention 
in the sequel. : 


16 The Phase Rule 


If, then, two variables, P and 7’, p and » or v and 7, be 
given, a system consisting of a single phase is fully deter- 
mined: if one only, there remains one choice before the 
determination. If none be given, then clearly two of the 
variable factors may be chosen at random (again within 
limits) before the system is adequately defined. But once 
two are selected, there is no further possibility of arbitrary 
choice: the third is settled and the system defined. Hence 
the maximum number of degrees of freedom possessed by 
a single phase of a homogeneous chemical substance is two. 

Two phases of one component. Instead of considering an 
isolated single phase, consider next two in equilibrium 
with one another, each of the same chemical composition. 
These may be solid and vapour, liquid and vapour, solid 
and liquid or liquid and liquid. It is a conclusion from 
experience that two vapours never coexist in separate 
phases: all gases and vapours mix with one another in 
every proportion. Nor is any case on record of the 
existence, in equilibrium, of two liquid phases of one 
component. 

Temperature and pressure must be the same in-any two 
coexisting phases if there be equilibrium, but the specific 
volumes will not necessarily be alike. In this case there 
will therefore be two equations connecting pressure, tem- 
perature and volume, the first holding for the one phase 
and the second for the other. For brevity, such equations 
will be referred to as ‘single-phase equations ’. 

Now, however, it becomes necessary to take into account 
yet another factor. Why does not the substance in one 
phase pass completely into the other, or vice versa? In 
short, why do the phases exist in equilibrium instead of 
amalgamating ? Presumably the equilibrium is not static, 
but dynamic, and as many molecules pass in one second 


The Phase Rule 17 


from one phase to the other as from the other to the one. 
There must, then, be some condition, no doubt a function 
of the variables in the respective phases (though the precise 
nature of it is quite immaterial to the argument), which 
determines the equality of the tendencies of the phases to 
pass into one another.* This condition may be expressed 
in what one may term, for convenience of reference, a 
‘ phase-equilibrium equation ’. 

Hence we have in the case of two coexisting phases of 
one component, three separate equations between the four 
variables, p, 7, v7, and v,. It follows that if one variable 
be fixed, or in other words, one degree of freedom be exer- 
cised, the values of the other variables are settled and the 
system fully defined. It is not possible arbitrarily to give 
values to more than one variable, and so it may be con- 
cluded that the maximum number of degrees of freedom 
possessed by a system containing in equilibrium two 
phases of one and thé same substance is one. 

Condition as to magnitude of phases. It might be well to 
emphasize the assumption (implicit in the . statement 
already made that capillary forces are taken as negligible) 
maintained throughout any discussion on the Phase Rule, 
namely, that so much of each phase is present in the 
system that the addition of more of it does not affect the 
equilibrium: or in other words, that the equilibrium is 
independent of the absolute or relative amounts of the two 
phases. Phis will be the case only when the phases are 
present in macroscopic masses; not when they are present 
in such minute amounts that the relative importance of the 
surface energy depends upon the state of division, as in fine 
drops of liquids or in crystals only a few microns in diameter. 

* In this connexion, see Chapter VII for a brief account of what is 
known as ‘ chemical potential ’. 

2568 B 


18 The Phase Rule 


Three phases of one component, Going further, consider a 
system of three phases in equilibrium. Again, temperature 
and pressure must be identical im all phases or obvious 
adjustments must occur: but specific volumes may differ 
so that there will be three equations (one for each phase) 
connecting the variables p, 7, v,, v, and v,. In addition, 
there will be an equilibrium equation between phases I and 
II, and another between phases I and III. Since two 
phases, each of which is in equilibrium with a third phase, 
must be in equilibrium with one another (a statement 
often referred to as the Law of the Mutual Compatibility of 
Phases), it follows that the equilibrium equation of phases 
II and III will not be independent of these two. 'There- 
fore, between the five variables there are five equations. 
The values of these variables are thus completely settled in 
terms of the constants of the equations, and no freedom is 
left for arbitrary assignment of specific values to any of 
them. Ina system of three phases of the same chemical 
substance there are not, then, any degrees of freedom. 
There is only one possible value for each of its five deter- 
mining factors, and this is expressible in terms of definite 
natural constants. 

Four phases of one component impossible. If four phases 
were in equilibrium, say two solids, liquid and vapour, it 
would follow from similar reasoning that between the six 
variables 7, 7, v,, v,, Vz and v,, there would be four distinct 
equations, one for each phase connecting its variables, and 
three phase-equilibrium equations ; in all, seven equations, 
quite independent. The six variables could be calculated 
from six of the equations and the values so found would 
not fit in with a seventh entirely independent relation. 
Hence the coexistence of four phases of one and the same 
substance is an impossibility. Three is the maximum figure, 


The Phase Rule 19 


It has now been seen that in systems of one component 
we have, for one phase, two degrees of freedom to select 
variables consistent with the continued stable existence of 
the phase; for two phases in equilibrium, one degree of 
freedom ; and for three phases, none. Also not more than 
three phases may coexist. These results may be expressed 
in the form 

F se — 3 
for any one-component, or unary, system, where P is the 
number of coexisting phases and / is the number of 
degrees of freedom possessed by the system. Hxamples 
will be discussed in Chapter IIT. 

Two-component systems. Ifa single component can exist 
in at least three different phases, there would appear to be 
a considerable increase in the number of possible systems 
of coexisting phases when a second component is intro- 
duced. But actually the number is not great. For one 
thing, there can never be, or at any rate there never is, 
more than one vapour phase: many liquids, too, are miscible 
in all proportions, while some others which might be only 
partially miscible, such as water and molten salt, may not 
both be able to exist liquefied in the same temperature 
range: some pairs of solids, too, are miscible and so may 
form one phase instead of two. But, besides these limita- 
tions there is another, similar to that already found in the 
ease of one component. 

One phase of two components. As the simplest case, 
assume that two components, say salt and water, form only 
one phase, a homogeneous solution. To define this solution 
it is necessary to state its temperature, the pressure exerted 
upon it, and the specific volume of each component. Instead 
of dealing further with specific volumes, there will now be 
adopted the more usual practice of taking concentrations. 

B 2 


20 The Phase Rale 


Specific volume is the volume occupied by unit mass; con- 
centration is mass in unit volume. One cubic centimetre 
(or any other selected unit of volume) will contain c, grammes 
of the first component and c, of the second ; and c, will not 
be expressible in terms of c, unless the density of the phase 
be known. Thus, of the three quantities ¢c,, ¢, and density, 
two will always be independently variable, and these will 
be taken to be ec, and «,. 

There are now four variables, », 7, c, and ¢c,, and only one 
equation, of the single-phase type, between them. Obviously 
then, three of the four variables may be fixed arbitrarily 
without violating the necessary condition of the equation 
which at once fixes the fourth in terms of these three. 
Where two components are concerned there are thus three 
degrees of freedom possessed by any system containing 
a single phase. 

More than one phase of two components, Similarly when 
two phases, each containing the two components, coexist, 
there will be two single-phase equations and one phase- 
equilibrium equation for each component: four equations 
in all between the six variables p, 7, ¢,, ¢,, ¢,/ ande. It 
follows that the freedom of selection of variables consistent 
with the continued coexistence of the two phases is limited to 
two: the remaining four are then determined in accordance 
with the equations quoted. Thus, the number of degrees of 
freedom possessed by a two-component system in two 
phases is two. 

With three phases there will be eight variables, p, 7, 
C13 Cas Cy’) Ca’, C,” and ¢,” (where dashes denote phases in an 
obvious manner), and seven equations, three being of the 
one type and four of the other. There remains, there- 
fore, only one degree of freedom where three phases 
coexist. 


The Phase Rule | 21 


So, too, where there are four phases, there will be ten 
variables, namely, p, Z’ and two concentrations in each of 
the four phases. Between the variables there will be four 
single-phase equations and six phase-equilibrium equations. 
So that every one of the variables will be expressible in 
terms of the natural constants of the equations, and no 
freedom for arbitrarily fixing any one of them will remain. 
There are, then, no degrees of freedom in a system con- 
taining two components in four phases. 

From the coexistence of five phases we should, by 
similar reasoning, obtain thirteen independent equations 
between twelve variables. This is impossible: hence the 
supposition of the existence of five phases in equilibrium 
cannot be maintained. Four is the maximum number. 

_ Putting all these facts together, it is clear that in two- 
component systems 2+P= 4. 

The term ‘component’. Now in any two-component 
system it is possible that besides the two components and 
homogeneous mixtures of them in the solid, liquid or 
vaporous state, there may exist definite chemical com- 
pounds which, being chemically distinct bodies, might 
reasonably be ranked as new or additional components. 
There is, as a matter of fact, no particular reason why they 
should not, except the very good reason of convenience and 
simplicity. It is obvious that the composition of a com- 
pound, say a salt hydrate, can be expressed in an equation 
involving definite quantities of its components only: and 
so for every equation involving concentrations of the former 
a precisely equivalent one in terms of the latter may be 
deduced. For a given system, in short, the difference 
between variable factors and equations connecting them 
will remain the same if a compound existing in it be con- 
sidered a component additional to its two constituents, for 


re The Phase Rule 


the number of variables and the number of equations will 
be increased equally. 

It is clearly an advantage to refrain from this quite 
useless increase of variables and equations and to restrict 
the number of each to a minimum, thus maintaining com- 
plete independence among the equations. This may be 
done by adopting the rule that no compound occurring 
in a system will be regarded as a new component unless its 
composition be such that it cannot be expressed in terms 
of a minimum number of components already selected. In 
Nernst’s words, one must take as the number of components 
in a system the minimum number of molecular species 
in terms of which the compositions of all phases in that - 
system may be quantitatively expressed. The selection of 
the components obviously leaves room for arbitrary decision: 
the selection of the number does not. 

In case there may be difficulty in deciding this really 
quite simple matter the following general rule will always 
give the information desired : 

From the total number of distinct chemical (mo- 
lecular) species in the several coexisting phases 
subtract the number of chemical equations required to 
express the changes which may occur when equi- 
librium is disturbed. ‘The remainder is the number 
of components in the sense desired. 

Example of determination of number of components. As an 
example which may be helpful to the student, take the 
system in which there may exist, according to cireum- 
stances, the following chemical species :—Calcium oxide 
(CaO), water (H,O), carbon dioxide (CO,), calcium car- 
bonate (CaCO,), calcium hydroxide [Ca(OH),], carbonic 
acid (H,CO,) and caleium bicarbonate [Ca(HCO,),|. Here 
there are seven distinct molecular species. Between them 


The Phase Rule 23 


we have the following equations, which suffice to express 
the changes which may occur when equilibrium is dis- 
turbed : 

CaO + H,O = Ca(OH), 

Poe) oo, = HCO, 

C20 + CO, = CaCO, 

CaCO, + H,CO,; = Ca(HCoO,),. 


The difference between the number of molecular species 
(seven) and the number of equations (four) is three. If 
one increase the number of species by including ions, then 
there will be an exactly equal increase in the equations. 
The system is therefore one of three components. 

Since for the expression of compositions of phases it is 
quite immaterial in terms of what particular substances - 
the expression is made (indeed, we often do not know 
exactly what molecular or ionic species are present) so long 
as all the matter in the phase is accounted for, it follows 
that any three species may be selected as the components 
provided that the composition of any one of them cannot 
be stated in terms of the other two. In the ease cited, the 
most natural choice would no doubt be calcium oxide, 
carbon dioxide and water; for it is at once clear that the 
other four species may be regarded as composed of these in 
pairs, or altogether, as the case may be. But the same 
holds for other choices. Thus, taking calcium carbonate, 
calcium hydroxide and calcium bicarbonate as components, 
the other possible constituents of phases may be expressed 

- in terms of them thus : 


H,CO, = Ca(HCO,),— CaCO, 
2CO, = Ca(HCO,),—Ca(OH), 

2H,O = Ca(OH), —2CaCO, + Ca(HCO,), 
2CaO = 2CaCO,+Ca(OH),—Ca(HCO,),. ’ 


24 The Phase Rule 


Generalization. Having now defined strictly all the 
terms that/will be used, and having considered in detail 
the various possibilities in the two simplest types of 
system, one may proceed to obtain a general statement of 
the relations in any system between number of phases, 
number of components, number of variables and number of 
independent relations connecting them, the difference 
between the two latter (if any) being the degrees of freedom 
of the system. 

Suppose that P phases exist in equilibrium with one 
another and that their compositions may all be expressed in 
terms of C components. For each phase there will be an 
equation connecting its variables, pressure, temperature 
and concentrations; that is to say, there will be P single- 
phase equations. Also, the P phases may be arranged in 
P—1 pairs, so that there will be P—1 phase-equilibrium 
equations for each component, or C (P—1) altogether. 

The total number of equations 1s thus P+C(P—1).* 

The variables will be pressure, temperature and the 
concentration of each component in each phase; alto- 
gether, Cx P concentrations. The total number of variables 
is thus 24 OP. 

Representing by / the difference between the number 
of variables and the number of equations connecting them, 
that is, the number of degrees of freedom of the system, 


we have i (2+CP)—-[P+C(P-1)] 
= 9—P4 C, 
Or F4+P=C+2. 


* It must be remembered that concentrations are assumed to be 
expressed in quantities of matter per unit volume. If an expression in 
fractions of an arbitrarily specified total mass be preferred, the argu- 
ment takes a slightly different form, though the conclusion is, of course, 
the same. Compare Partington, Proc. Chem. Soc.,1911, p. 13. 


The Phase Rule 25 


This very simple relation is what is termed the Phase 
Rule. The conclusions already obtained in the earlier 
consideration of certain simple systems, namely, 7+ P= 38 
and +P = 4, are at once seen to be the special cases in 
which C=1 and C= 2, respectively. 

Nomenclature. It is convenient when referring to sys- 
tems to adopt a nomenclature as brief as possible, and it 
has become customary to describe any system first in terms 
of the number of components present, and then in terms of 
the number of degrees of freedom possessed by it. 

Thus, for systems of one, two, three, four, five (and so on) 
components, one uses the terms unary, binary, ternary, 
quaternary, quinary (and so on), respectively. When 
/’=0 the system is said to be invariant, since none of its 
variables may be altered at will without destroying the 
system in the sense of altering the number of coexisting 
phases. If / = 1, the system is univariant; and we have, 
in order, the terms bivariant, tervariant, quadrivariant, 
quinvariant and so forth. 


CHAPTER II 


THE PRINCIPLE OF LE CHATELIER AND 
BRAUN 


Ir a system in equilibrium be subjected to an alteration 
in one of the factors conditioning the equilibrium, it is 
clearly a matter of importance to be able to state, qualita- 
tively at any rate, in what direction displacement will 
occur if and when the system attempts to accommodate 
itself to the new condition. A generalization making this 


26 The Principle of 


possible was put forward in 1885 by Le Chatelier, and 
independently in the following year by F. Braun. It was 
based solely on experience and has proved to be quite 
general in its application, being in fact covered by the 
Second Law of Thermodynamics. 

Illustrations of the principle. The nature of the law is 
best seen from specific examples. Suppose that a system 
in equilibrium, say liquid water and water vapour, in a 
fixed space and at a certain temperature and pressure, be 
supplied with a definite amount of heat. If no change 
take place in the system there will be a rise in temperature, 
dependent only upon the quantities and specific heats of the 
liquid and vapour. If, however, change occur sponta- 
neously and a new equilibrium condition be attained, it 
will be found that the rise in temperature is less. The 
consequence of adding heat, namely, rise of temperature, 
has been Jessened by the process of adjustment of equili- 
brium between the phases. In this particular case some 
of the heat has been used to cause evaporation of a certain 
amount of water, the vapour pressure of which is thereby 
increased : the remainder alone has been available to in- 
crease temperature. 

Similarly, if the volume available to the two phases at 
a fixed temperature be reduced to a specified extent, there 
will be, if the phases remain otherwise unaltered, an in- 
crease in the pressure exerted by the system. But, if 
change occur and a new equilibrium be attained with dif- 
ferent relative amounts of liquid and vapour, the increase 
in pressure will be less. The alteration of the relative 
amounts must therefore have been one involving decrease 
in volume, that is, condensation of vapour. 

Or again, if solid and liquid be in equilibrium in a 
certain space and the volume of the system be reduced 


Le Chatelier and Braun 27 


a fraction by compression, the resultant pressure will be 
greater when the relative amounts of each phase remain 
constant than it will be if these amounts adjust themselves 
naturally. The lower pressure in the latter case can only 
be the result of an adjustment involving a decrease in 
volume. If on melting the solid expands then more solid 
must have been formed: if it contracts then some must 
have melted. | 

General statement of principle. In all three cases we 
have the same general rule. Given a change in the con- 
ditions under which a system is in equilibrium, the conse- 
quent alteration in the factors which specify the system 
will always be less for a change to a new state of equili- 
brium than for a change in which the system remains 
otherwise as before. 

It is essential, however, to be more precise in speaking 
of changes and alterations in the factors of a system so 
that cases may not occur in which there is ambiguity. 
When, for example, a gaseous system is subjected to 
adiabatic compression, its temperature is raised; that is to 
say, the translational energy of its molecules is increased, 
and therefore the pressure exerted by them is also in- 
creased. Hence the spontaneous change in the system, 
development of heat, has tended to intensify the initial 
increase of pressure, in apparent contradiction to the 
general statement just made. It therefore would seem 
that some more precise formulation of the principle 1s 
required: pressure and temperature are evidently not the 
factors to be considered in a case such as this. 

This formulation may best be arrived at by emphasizing 
the distinction between intensity and quantity factors 
respectively. Pressure, for example, is an intensity factor ; 
volume is the corresponding quantity factor. Temperature 


28 The Principle of 


and entropy stand in a similar relation; so do electro- 
motive force and electric current, surface tension and 
surface, osmotic pressure and volume occupied by unit 
mass of the solute (dilution). The variables temperature 
and pressure which enter so largely into the considerations 
which follow are thus both intensity factors. 

Now the product of the intensity factor and the corre- 
sponding quantity factor is always of the dimensions of 
energy, and if changes be brought about in the quantity ~ 
factors, say, and be represented by dq,, dqo,+..d¢,, the 
total change of energy dH will be given by 


dH = L,dq,+L,dqgt+...+lL,4n 


where J,, /,,... /,, are the corresponding intensity factors. 

When, then, systems are subjected to stresses or strains 
of any kind, it is to the relations between corresponding’ 
intensity and quantity factors that attention must be 
directed. 

Precise statement of principle. The principle may now 
be expressed definitely and without possibility of ambiguity 
in the following way: 

Any alteration in an intensity factor J, consequent 
upon a given alteration dg in the corresponding quan- 
tity factor, will be smaller for a change to a second 
state of equilibrium (with consequent adjustments of 
other factors) than it would be if all the other quantity 
factors were to remain constant. 

The same thing is, of course, intended to be conveyed 
by the more usual statements of the rule, as, for example, 
that every system in equilibrium is conservative; that a 
system tends to change in such a way as to oppose and 
partially to annul any alteration that may occur in a factor ; 
that if a system in equilibrium be subjected to a constraint 


Le Chatelier and Braun — 29 


by which the equilibrium is shifted, a reaction takes place 
which opposes the constraint. Such statements, however, 
are certainly not sufficiently specific.  - 

The principle of Le Chatelier and Braun is only quali- 
tative, and thus enables us to predict only the direction in 
which the variables of a system will change when one of 
them is altered. By the aid of the laws of thermody- 
namics quantitative expressions embodying it may be 
deduced: but for the present it suffices to retain the quali- 
tative result. Instances in accordance with the rule will 
occur repeatedly in the sequel. 

For a more adequate discussion the student is referred to 
papers by Ehrenfest (Zeit. phys. Chem., 1911, 77, 2) and 
Lord Rayleigh (Journ. Chem. Soc., 1917, 111, 250), 


CHAPTER III 


ONE-COMPONENT SYSTEMS 


Tur Phase Rule, as deduced in Chapter I, deals with the 
number of factors concerned in defining equilibrium, and 
it is apparent that in studying the application of the rule 
to specific cases we shall be dealing primarily with the 
interdependence of these factors. In any one-component 
system the number of factors, or variables as one may 
continue to call them, the values of which determine the 
equilibrium, is three, these being temperature (7’), pres- 
sure (p) and specific volume (v). The latter, volume in 
which unit mass is contained, may of course be replaced by 
density, or mass present in unit volume. 


30 One-Component Systems 


To show the interdependence of the three variables in 
a given one-component system a graphical representation 
in three dimensions will be necessary, and the immediate 
consideration of a typical space model is probably the best 
introduction to the particular cases to be studied.* 

A typical 3-dimensional model. It is not possible to give 
one general model which will represent rigorously the 
relations of 7, v and 7’ in all one-component systems, since 
the relative positions of some of the regions mapped out 
vary according to certain characteristic properties of the 
component. But the alterations involved are readily made ; 
and Fieure 1 (taken from Janecke, Geséittigte Salzlisungen, 
p- 11), which represents in perspective, quite diagrammati- 
eally, y, v and 7’ for a one-component system such as that 
of water, where at the freezing-point Vapour > “solid > 
“liquids May be taken to be typical, though it will require 
an obvious adjustment to make it applicable to the more 
frequent case, such as that of alcohol, where Yann > 
“liquid 7 “solids 

The three-phase system, As a starting-place in deserip- 
tion of the model, it is convenient to take the points a, 
6 and e which give the specifie volumes of liquid, solid and 
vapour, respectively, at the particular temperature and 
pressure at which they coexist. The line ade is there- 
fore at right angles to both pressure and temperature 

g S Pp p 

* This would appear not to be the general view of teachers writing 
on the subject. It is customary to deal only with plane diagrams, i.e. 
sections and projections, showing relations between two only of the 
three possible variables, the two most favoured in one-component 
systems being temperature and pressure. Such figures will be dealt 
with here in their turn, but it is considered that from a didactic point 
of view it is a distinct advantage to start from the complete model, and 
to obtain from it the partial representations of the various possible 


two-dimensional figures. A similar course will be followed in the 
higher systems. 


One-Component Systems 31 


axes. It is needless to say that in any particular case, 
such as that of water, the length ad would be very much 
less, relatively to ac, than it is shown to be in the figure. 










limes oe we wmeonces & ee were wee ee est ee Sete se oem ee 








ew ee ee ee ee ew ere em ee 
ww wwe em ow 6 eee wee eH ee eo Heme 





Pressure —> 


ed 






le wewe ewe em oe wee ee ee we ewer ee eee ee em eee eee 
















Specific Vol.— 


Figure 1, 


Actually, for water, a corresponds with a value 7 = 
1-000132 cubie centimetres; 4 with v = 1-0917 ; ¢ with 7 
= 210 litres. The whole model has been distorted in order 
to bring out with sufficient clearness its different divisions. 


32 One-Component Systems 


Since along the line adc there are three phases coexis- 
tent, it follows from the phase rule, since C= 1, that F=0. 
One may not, therefore, alter any of the conditioning 
factors. The system is invariant. To attempt to effect 
any alteration would result in the disappearance of one of 
the phases. Attempted pressure increase would cause the 
disappearance of the vapour phase; persistent attempt to 
raise the temperature would result in the solid disappearing ; 
and so on. ‘The pressure and temperature of coexistence 
of solid, liquid and vapour in a one-component system are 
single-valued constants of nature, and so are the respective 
specific volumes. 

In the water system the value of the pressure is 4-4 
millimetres of mercury; that of the temperature is 
+0-0076° C.; the volumes have been quoted above. 

The term ‘pressure’, It is necessary to be quite clear 
about the meaning attaching to the term ‘pressure’ in 
such a case as this. It must be understood to mean that 
pressure which the vapour phase (and the other phases) of 
the component exerts, and is on no account to be confused 
with such a total pressure as may be exerted through, say, 
the agency of some other substance, or mixture of sub- 
stances, such as air. Unless this be borne in mind there 
will be difficulty in reconciling: the invariance of the three- 
phase water system with the facts underlying the com- 
monly quoted but inaccurate statement that the three 
phases, ice, liquid water and water vapour, may coexist 
at other temperatures if confined in a vessel also containing 
air or other gas, the pressure upon which is varied. ‘Thus, 
it. is often said that these three phases coexist at 0° C, 
under a total pressure, exerted partly by the water vapour 
and partly by the atmosphere, equal to 760 mm. of mercury ; 
or at —0-059° C. under 8-1 atmospheres, or —0-129° under” 


One-Component Systems 38 


16-8 atmospheres. Actually they do nothing of the 
kind. 

For if one considers the equilibrium relations of the 
system abc of Figure 1, one is led to the conclusion that 
the only possible way of altering the pressure on the 
system is by introducing a new component; most simply, 
a gas. There is no other way of changing the pressure 
from 4-6 mm. while maintaining the coexistence of three 
phases. In the simplest case, where the new component is 
a pure gas, the system will, however, at once become one 
of two components, or binary: if the gas be a mixture, the 
system may, and usually will, become of a still higher 
order. Corresponding with the increase of components 
there will be an increase in degrees of freedom, and at once 
it becomes apparent that three phases may be in equilibrium 
at other values of the variables than those represented at 
a,dandc. These phases, though, will not be ice, liquid 
water and water vapour; they will be ice saturated. with 
the new gaseous component (for that solids may dissolve 
eases, however slightly, can hardly be denied), liquid water 
also saturated with it, and a vapour phase which will be 
a mixture of water vapour and the gas. These are three 
phases belonging to a two-component system, having there- 
fore one degree of freedom, and are quite different from the 
three phases of the invariant one-component system, 

One cannot neglect the gas as a component, terming if 
‘neutral’, for while its solubility in both solid and liquid 
phases may be so slight as to seem quite negligible and to 
justify the term ‘neutral’, at any rate from a practical 
standpoint, its ‘ solubility ’ in the third phase, vapour, will 
of course be infinite. 

The two-phase system: liquid and vapour. Suppose now 
that in the three-phase system represented at a, 6 and c in 


2568 C 


34 One-Component Systems 


Figure 1, an attempt be made to raise the temperature of 
the three phases by applying heat. It will be found that 
the ice will melt, for experience has shown that under no 
circumstances yet discovered may ice be heated above 
0:0076° C. in presence of both liquid and vapour. A transfer 
of matter from one phase to another, or a phase reaction, 
will take place. In melting, the ice will absorb heat, and 
if there be sufficient ice and only a limited supply of heat, 
the attempt to alter the temperature of the system may 
fail to do more than change the proportions of ice and 
liquid. This is a case with which Le Chatelier’s theorem 
is clearly in accord. If, however, the application of heat 
be persisted in, all the ice will melt before the temperature 
begins to rise, and only when liquid and vapour are the 
sole phases remaining can a value above 0-0076° be 
reached, : 

At the moment when the whole of the ice has just dis- 
appeared the system is represented by the two points a and 
ce of Figure 1. It is a one-component system in two 
phases, so that /= 1. Without altering the phase equi- 
librium one may (within limits) arbitrarily vary one con- 
dition of the system, either pressure or temperature. This 
of course includes alteration of specific volume, since there 
is no other way of doing this than by altering pressure or 
temperature. But when one freedom of choice has been 
exercised, say a particular temperature selected, the remain- 
ing conditions, p and v, adjust themselves accordingly and — 
cannot be varied without bringing about the disappearance 
of one of the phases. . 

Suppose that the temper care be that corresponding with 
the line yz. The vapour pressure will increase to a par- 
ticular value which may be read from the model. The 
specific volume of the liquid will increase or decrease 


One-Component Systems 7 35 


according to whether the expansion due to rise of tempera- 
ture be greater or less than the contraction due to increase 
of pressure. It is probably an invariable rule that the 
coefficient of expansion by heat is greater than the com- 
pressibility due to corresponding alteration of vapour 
pressure, so that the volume at y will be greater than that 
at a. On the other hand, the specific volume of the 
saturated vapour at the higher temperature will be less 
than that at the lower, for the effect of increased tempera- 
ture in expanding the vapour is far more than compensated 
by the greater amount of vapour per unit volume required 
for equilibrium with the liquid. The length of yz is there- 
~fore less than that of ac, or, in other words, with rise of 
temperature the specific volumes of vapour and_ liquid 
approach one another. At a sufficiently high temperature 
(and pressure) they may be expected to become equal, and 
such a point is shown in the diagram at d. 

The critical pont. One cannot say that at this point the 
one or the other phase has disappeared, but simply that the 
two phases have become indistinguishable from one another. 
Each contains the same amount of the same component in 
the same state in each cubic centimetre, and this implies 
identity in all respects. This point is always known as 
the critical point, and the three factors as critical pressure, 
eritical temperature and critical volume, respectively. For 
the water system, the approximate values are 195 atmo- 
spheres, 360°C. and (for one gramme) 4-8 cubic centi- 
metres. 

Limits of liquid-vapour system. The continuous curve 
aydzc gives all the values which p, 7’ and v (or rather, v, 
and v,) may assume consistently with the continued co- 
existence of liquid and vapour. If the endeavour be made 


to subject the two-phase system to a pressure or temperature 
C2 


36 One-Component Systems 


not corresponding with this bounding line, one phase 
or other will disappear; and that is why, when it was 
stated that the vapour-liquid system possessed one degree 
of freedom, care was taken to make it clear that this 
freedom was not infinite, but that its exercise was restricted 
to certain limits. The curve aydzc gives the precise 
definition of these limits in the present case. 

Any point in the area bounded by this curve and the 
straight line ae will give an average specific volume of two 
phases coexisting in certain proportions. Thus the point 
x gives for a particular temperature and pressure the mean 
specific volume of a mixture of liquid and vapour in which 
the fraction #z/yz is liquid of specific volume corresponding 
with the point y, and ay/yz is vapour of specific volume 
corresponding with the point z. The area aydzc is thus 
the field of coexistence of liquid and vapour, or the area ot 
the univariant system. 

The two-phase system: solid and liquid. Returning to the 
line adc of the invariant system, assume this system to be 
subjected to persistent infinitesimally increased pressure on 
the vapour, instead of increased heat-supply. Vapour will 
condense in the endeavour to reach the lower equilibrium 
_ pressure and escape the restraint, and ultimately it will 
disappear as a separate phase. ‘There will thus be none of 
the component remaining in the condition corresponding 
with the point c, and the other two phases of liquid and 
solid will be represented by a and 4, Again, therefore, the 
attempt to exercise a degree of freedom upon the three- 
phase system has produced a two-phase system, and this 
may be subjected to one factor arbitrarily chosen (within 
limits) without further alteration in the number of phases. 

The conditions governing the equilibrium between solid 
and liquid have been examined in detail for only a few sub- 


One-Component Systems 37 


stances. The specific volumes of both phases will in general 
be decreased by increase of pressure, that of the liquid more 
so than that of the solid. But whether increase of pressure 
will raise or lower the temperature of coexistence of the 
two, phases, conveniently called the melting-point, depends 
upon the relative values of the specific volumes. When the 
solid, on melting, contracts, increase of pressure will, in 
accordance with the rule of Le Chatelier and Braun, bring 
about that change, namely, diminution of volume, which 
tends to relieve the pressure. The solid will therefore 
liquefy more readily, that is, at a lower temperature. If, 
on the other hand, the solid expand on melting, increase of 
pressure will raise the melting-point. The latter case is by 
far the commoner: the former holds for a comparatively 
small number of substances, including water, potassium 
nitrate, bismuth and cast iron. 

The precise directions of slope of the curves be and af will 
therefore depend upon the nature of the component. There 
is not necessarily any limit to the course of these curves 
with increasing pressure, except the limit imposed by diffi- 
culties of experiment. The area abef may then be regarded 
as a portion of that of the two-phase system containing 
liquid water and ice. 

Improhability of solid-liquid critical point. There has been 
much interesting speculation as to whether a critical point 
be possible between solid and liquid phases similar to that 
at point d in Figure 1 between liquid and vapour. With- 
out going into details it may be pointed out that there is 
a fundamental difference in the two cases which rather 
weakens any argument from the one to the other by analogy. 
In both liquid and vapour phases the molecules are without 
order in arrangement. The phases differ in the average dis- 
tances between molecules and in other properties dependent 


38 One-Component Systems 


thereon, so that a continuous passage from one state to the 
other is readily conceivable. 

In a solid, on the otber hand, the molecules are arranged 
in definite geometrical order (if indeed one may regard the 
solid structure as composed of separate molecules at all), and 
to picture a continuous passage from that to the disorder of 
a liquid is a little difficult. The possibility of a solid-liquid - 
critical point is, to say the least of it, rather doubtful. 

The two-phase system: solid and vapour. One other change 
from the three-phase system adc to a two-phase system is 
possible. If an attempt be made to lower the temperature, 
water will solidify in the attempt to restore the invariant 
conditions : but persistent removal of heat will result in the 
complete disappearance of liquid. At the moment of its 
disappearance, if the cooling has been infinitely slow, the 
system will be represented by the two points 0 and ¢ for 
solid and vapour respectively. This system has one degree 
of freedom, and temperature or pressure may be varied 
arbitrarily, within limits, without destroying it. On lower- 
ing the temperature, the specific volume of the ice will 
steadily decrease along the curve lg, though only slightly. 

The equilibrium vapour pressure will decrease and the 
specific volume of the vapour will increase as shown by the 
curve ch, The two curves dg and ch therefore diverge from 
each other with decrease of temperature. The decrease of 
vapour pressure with temperature in the region of the 
point ¢ is greater per degree for ice than for water: hence 
the curve c/, which is de produced, would lie above the 
curve cz, and conversely ze produced would le above cf. 
The lower limits of the curves 4g and ch will be reached 
only at the absolute zero of temperature: the vapour 
pressure will then be zero, which means that the specific 
volume of the vapour approaches infinity as the zero tem- 


One-Component Systems 39 


perature is reached. The volume of the solid will have a 
definite value. 

Within the region bounded by dc and the curves ég and 
ch, or their continuations, the conditions are such that the 
one component must always exist in two phases, solid and 
vapour. Any attempt to impose conditions corresponding 
with points outside this area will effect the change of one 
phase into the other, giving a single-phase system with two 
degrees of freedom. 


Pressure —> 





Specific Vol.—> 


FIGurE 2. 


Single-phase systems. There are three such single-phase 
systems, the relative positions of which are shown in 
Figure 1 by the areas marked solid, liquid and vapour, 
respectively. ‘The liquid and vapour areas merge into one 
another above the critical point: but there is no evidence 
in the case of water of a similar merging: of ice and vapour 
areas or.of ice and liquid areas. A critical point (unstable) 
between ice and vapour is improbable for the same reasons 
that make one between ice and liquid seem unlikely. 

To turn Figure 1 into a diagrammatic representation of 


a one-component system in which %,.1;4 < jquid<Vvapour 


40 One-Component Systems 


only a slight adjustment of the relative positions of the 
areas is required, and Figure 2 indicates the change required. 

Summary. Summarizing what has been said of Figure 1 
(and, mutatis mutandis, the same applies to Figure 2), all 
realizable equilibrium associations of the variables 7, p and 
v may be represented for a one-component system on certain 
surfaces and their intersections at lines or points. Any 
values of », 7’ and v represented by points not lying on 
these surfaces or their intersections cannot belong to 
a stable phase or to phases in stable equilibrium. Meta- 
stable equilibria are possible in some cases, but the present _ 
discussion is not concerned with them. 

Of the areas, those marked solid, liquid and vapour, 
respectively, are systems of a single phase and bivariant: 
those marked solid and liquid, solid and vapour, and liquid 
and vapour are systems of two phases and univariant. The 
latter areas differ from the former in that they are plane in 
one direction, the ends of lines (which may be termed con- 
jugation lines) in this direction giving the values of the 
variables for the respective coexisting phases. Finally, the 
points a, 0 and ¢ represent together a three-phase or invariant 
system. 

Projection onp-T plane. Now consider this three-dimen- 
sional figure from certain more restricted points of view. 
One of these leads to the diagram, reproduced in Figure 3, 
which is usually given to represent the one-component 
system water. 

Pressures are plotted on the axis of ordinates and tempera- 
tures on that of abscissae. If this figure be considered in 
relation to Figure 1 it will be seen that it is simply the 
orthogonal projection of the space model on the pressure- 
temperature wall. This comes to the same thing as regard- 
ing it as Figure 1 in which the volume axis has been taken 


One-Component Systems 4] 


with such huge volumes corresponding with unit length, 
that differences in volume of the order dealt with are beyond 
the possibility of representation as finite distances. 

The three points a, ) and ¢ now become one at 4, and 
hence to this is always given the name of ‘triple point’. 
Points such as y and z, e and /, g and 4, fall together at J, 
£ and G, respectively. The line of the three-phase system 
has become a point: the areas (plane in one direction) of 


Pressure —- 





Temperature —> 
FI@uRE 3. 


the two-phase systems have become lines: the areas of the 
single-phase systems remain areas, but are now plane. 

The curve for the liquid and vapour system begins at 4 
and ends sharply at d@, the critical point. ‘The solid-vapour 
curve AG will end at zero pressure and zero (absolute) 
temperature. ‘The end of the solid-liquid curve 4/ cannot 
be predicted. The complications in it in the case of water 
are shown in their pressure-temperature relations in Figure 
4, where the division of the field, so far as investigated, 


42 One-Component Systems 


between liquid and the various solids, Ice I, Ice II, Ice IL, 
Ice V and Ice VI, is clear. In interpretation, this figure 
offers nothing new as compared with Figure 3. 

The slope of the curve 47 in Figures 3 and 4 towards 
the pressure axis is in accordance with Le Chatelier’s 
theorem and the fact that ordinary ice (or-Ice I), on melting, 
contracts in volume. If the volume change were in the 
opposite direction the curve would slope away from the 
pressure axis. Hxam- 
ples of this are seen in 
the curves BD, DF and 
FH of Figure 4. 

Phase relations at at- 
mospheric pressure. One 
small point frequently 
appearing in practice 
may be illustrated from 
Figure 3. The value of 
pat the triple point abe 

050°C ae 40 will be different for 

Temperature —> different substances. It 

may be, and for most 

common liquids like water is, below the atmospheric pressure 

of 760 mm. of mercury represented by the horizontal line 

PP. It is clearly possible in any such case to obtain the 

three phases, in pairs or singly, in systems of varying 

temperature open to the atmosphere. Thus, on applying 
heat to the solid, it melts at J/ and the liquid boils at B. 

But there are other cases, exemplified by ammonium 
chloride and carbon dioxide, where the 760 mm. pressure 
line, represented now by /’P’ in Figure 3, lies below the 
triple point, cutting only the vapour-solid curve Gd. In 
such cases, liquid cannot be obtained in equilibrium experi- 





8 


Pressure (Kilos.x 10/em’) 


FigurE 4. 


One-Component Systems 


43 


ments unless means be taken to raise the pressure above 
The only possible change for the 


that of the atmosphere. 


solid, if heated, is to 
vaporize at S: con- 
versely, vapour, if 
cooled, condenses di- 
rectly to solid. These 
phenomena, in ~ se- 
quence, make up the 
process of sublimation. 


Projections on v-T 


and p-V planes. Re- 
turning again to 


Figure 1, it may be 
seen that two other 
two-dimensional pro- 
jections, besides Fig- 
ure 3,may be obtained, 
namely, those on the 


_ volume - temperature. 


and the volume-pres- 
sure planes, respec- 
tively. The former is 
shown diagrammati- 
cally in Figure 5. 
The region abef of 
coexistence of solid 
and liquid is very small 
on account of the 
slight upward slopes 





‘Temperature —> 


Pressure —> 


of the curves Je and af. 










ff —- 
ens 


LIQUID 
and VAPOUR 


VAPOUR 














SOLID ; 
SOLID and VAPOUR 
Specific Vol. ——>» 
FIGURE 5. 


and VAPOUR 








Specific Vol. = 


FIGURE 6. 


This area will become closed 
similarly to that of abcd for liquid and vapour only in the 
unlikely event of there being a liquid-solid critical point a. 


44 ‘One-Component Systems 


The remaining projection is given in Figure 6. An 
actual example of this, commonly quoted in text-books, is 
Andrews’s pressure-volume diagram for carbon dioxide. 

This projection involves the partial superposition of the 
solid-liquid and liquid-vapour fields. For the rest, both it 
and Figure 5 should be readily intelligible if the discussion 
of Figure 1 has been followed. A few isothermal lines have 
been drawn in Figure 6: similarly isobars might be drawn 
in Figure 5 and isochors in Figure 3. 

Possibility of additional phases. Although no single 
component can exist in two distinct vaporous forms consti- 
tuting separate phases, it is quite possible for two or more 
distinct solid, or even liquid,* phases to appear. In such 
cases there will be additional divisions in the p, 7’, v model 
to represent the conditions in which the added possible one-, 
two- and three-phase systems may exist. There will not, 
however, be anything in such systems essentially different 
from what has already been sketched in Figure 1. If, for 
example, a component may exist in four phases, say two 
different solids, liquid and vapour, represented respectively 
by S,, 8,, L and /, then there may exist according to con- 
ditions four one-phase systems, six two-phase systems, viz. 
S,—S8,, 8, —L, 8,—L, 8,—-V, §,-V and L—YF, and four 
three-phase systems, viz. S,—S,—L, 8,-L—V, 8,-L—V | 
and S,—S,—V.. But there will not be a system 
S,—S,—L—V in which all four phases coexist, for such 
a system would require, in accordance with the Phase Rule, 
F+P=C+2, that there be a negative degree of freedom, 
and that is meaningless. 

Hence further cases of one-component systems introduce 
nothing more than repetition of areas, lines and points 


* Asa matter of fact no case has yet been found of a single com- 
ponent existing in two distinct isotropic liquid phases. 


One-Component Systems 45 


similar to those of Figure 1. It will therefore suffice to 
indicate by means of projection diagrams the complications 
introduced, and for this the most convenient is that on the 
pressure-temperature plane. 

One component giving 8,, 8,, Land V. Figure 7 repre- 
sents diagrammatically the one-component system of 
sulphur, which substance, besides giving liquid and vapour 


Pressure —> 





emperature — > 


Figure 7. 


phases, may exist according to conditions either as a 
rhombic or a monoclinic solid.* 
The curve AT’, is that of the two-phase system rhombic 


* Many complications appear in an experimental study of sulphur, 
owing to its marked capacity for forming molecules of varying com- 
plexity and to the fact that equilibrium between these is not always 
attained rapidly. Unless an investigator be quite certain that he has 
attained equilibrium, he is not justified in regarding the system as 
unary: it becomes binary or of a still higher order. The use of the 
term ‘ pseudo-binary’ in this connexion is now common. Many other 
one-component systems offer the same difficulties experimentally as 
does sulphur. 


46 One-Component Systems 


solid and vapour. Ifa second form did not exist, one would 
expect this curve to continue to a triple point 7, at which 
liquid would appear and from which two curves 7, B and 
7,C would run, representing liquid-vapour and rhombic- 
liquid systems, respectively. But actually, if perfectly stable 
systems alone be considered, it is not possible to reach the 
point 7, for between it and A, at 7, (about 95-5° C.), the 
rhombie crystals change to monoclinic. 7’, is a triple point 
between two solids and vapour. From it there proceed the 
two curves 7’,7, and 7',7,: the latter is the vapour pressure 
curve of monoclinic sulphur: the former gives the equilibria 
between the: two solid forms, and the fact that it slopes 
away from the pressure axis shows that when rhombic 
changes to monoclinic there is an increase in volume. 
Increase in pressure is thus unfavourable to the change in 
this direction and a higher temperature is necessary to 
bring it about. | 

At 7, (120° C.) monoclinic sulphur melts. 7, is the 
triple point where solid, liquid and vapour coexist. 7, Bis 
a normal vapour-pressure curve of a liquid and will end at 
a critical point. 7',7', isa solid-liquid curve similar to that 
of ice-water, with the difference that it slopes away from the 
pressure axis, monoclinic sulphur being denser than liquid 
sulphur. The slope of 7’,7', away from the pressure axis is 
less than that of 7,7’, and the two curves therefore meet at 
a point 7, (151° C. and 1,288 atmospheres pressure, and 
therefore much higher than shown in the diagram), Here 
three phases, two solid and one liquid, coexist. Beyond it 
the curve 7’,C is that of the two-phase system rhombic 
sulphur and liquid, and from its slope it is apparent that 
this solid is also specifically heavier than the liquid. 

The region in which monoclinic sulphur may exist is 
bounded by the three curves 7,73, 7,7, and 7,7... The 


One-Component Systems 47 


regions of existence of the remaining phases are as shown 
in the figure. | 
It is clear that the curves for the two-phase systems 
S,—4S, and S,—Z, though both sloping away from the 
pressure axis towards 7',, might be at such an angle that 
they would continuously diverge with increase of pres- 
sure instead of meeting at a triple point for S$,—S,—L: 
or they might slope, the one towards, and the other from, 





% 


Seen 

= 

ea ng 
jenn 





(LIQUID, CRYSTALS) 





wteen ce. 





> 
Temperature --——> K; 


Fiaure 8. 





Pressure —>» 








the axis. In the latter case the diagram would take the 
form of Figure 8. 

In such a case, unless the directions of 7,C and T', D 
change at higher pressures, it is not possible for the three 
phases S|, 8, and / ever to coexist, or for a stable two- 
phase system S, — Z to be realized. 

One component giving 8, liquid erystals, L and V. An 
instance somewhat similar, and which will serve as an 
example of a one-component system in which two liquid 
phases (one of which is not isotropic) as well as the solid 


48 One-Component Systenrs 


and vapour are found, is that of p-azoxyanisol. The field 
S, is in this case that of the so-called ‘liquid crystals’ (a 
term much more justifiable than ‘ crystalline liquid ’, which 
is occasionally used); and the curve 7',C slopes like 7D 
away from the pressure axis, but at a smallerangle. Hence 
here too there is the probability that the solid and (true or 
isotropic) liquid fields never become contiguous, or that 
solid p-azoxyanisol and its clear liquid form are unable to 
exist stably together. 


Pressure —~> 








Temperature —> 


Fiaure 9. 


One component giving several solids, Asa last example of 
a system of one component which may exist in several 
phases, that of ammonium nitrate, represented in Figure 9, 
may be quoted. 

Four different solid phases numbered 1, 2, 3 and 4, 
respectively, are possible according to conditions of tempera- 
ture and pressure, and the system is of especial interest in 
showing the existence of a triple point (a) where the three 
phases are all solid. 7, 7,, 7, and 7’, are all triple points: 
at each of the first three two solids and a gas are in 


One-Component Systems 49 


equilibrium: in the last, liquid, solid and gas. The 
directions of the curves 7a and 7',a were established by 
experiments going to 250 atmospheres, and on the basis of 
these Roozeboom predicted that there would be a triple 
point (with three solids in equilibrium) at about 69° C. and 
1,170 atmospheres. Actually, Tammann found it later to 
be at 642° C. and 930 atmospheres. From it another 
curve ab, also verified by 'ammann, gives the equilibria 
between the two solids numbered 2 and 4. The slopes of 
Tc and 7',d make it probable that no solid but that numbered | 
1 can ever exist in stable equilibrium with liquid ammonium 
nitrate. 

Unstable systems. So far attention has been directed 
only to the stable conditions of one-component systems. 
These are the conditions where, in thermodynamic terms, 
entropy is a maximum, or thermodynamic potential or 
internal energy a minimum, according to the particular 
type of the system. (See Chapter VII.) But it is often 
found that systems may exist in unstable equilibrium, the 
necessary condition being complete absence of the phase or 
phases of greater stability. Such cases are to be dis- 
tinguished from those of merely apparent equilibrium 
where conditions of temperature are such that the rate of 
change from unstable to stable forms is infinitesimal in the 
' presence of the latter. 

It is unnecessary to discuss such cases in any detail, but 


— two terms used in connexion with them must be mentioned. 


These terms serve to distinguish two types of system where 

a component is capable of existing in more forms than one. 
Enantiotropy. Tf each form be stable over some definite 

range of temperature and pressure conditions, and there be 

therefore some condition at which the two coexist in stable 

equilibrium, the phenomenon is known as enantiotropy 
2568 D 


50 One-Component Systems 


(Gk. évavrios, opposite ; tpom, turn), and the component 
is said to be enantiotropic. An example is given by 
sulphur and represented in Figure 7. 

Monotropy. If, however, one of the forms be always 
stable, the other unstable, however pressure and temperature 
may vary, the phenomenon is termed monotropy (Gk. povos, 
single), and the substance is called monotropie. Figure 10 
illustrates monotropy. 


Pressure —> 





Temperature —> 


FieureE 10. 


AF is the vapour pressure curve of the stable solid form, 
SC of the liquid. DHF is the curve of the unstable solid 
form. It lies above 4B and does not at any point cut it. 
The two forms cannot anywhere exist in stable equilibrium. 
A melting-point of the second is possible at #, and a 
transition point with the first: form at /,-but in both cases 
the systems are labile or unstable, and at equilibrium 
change to solid and liquid respectively. 

Any one-component system may now be interpreted by 
reference to the typical cases which have been discussed. 


> 


51 


CHAPTER IV 


TWO-COMPONENT SYSTEMS 


’ General. The introduction of a second component adds 
very much to the possible complication of heterogeneous 
equilibria. The maximum number of phases which may 
coexist is four, a system containing these being invariant. 
The apparently possible complexity is, however, not 
realized in practice. ‘Thus, since all gases are completely 
miscible with one another, there is never more than one 
vapour phase ; and no case is known where more than two 
liquid phases coexist, though there is not any reason on 
theoretical grounds to suppose the coexistence of three or 
even four liquids to be impossible; but the number of 
solids (single components or compounds of them) in equili- 
brium may reach the maximum of four. 

In a univariant system there will be three phases ; in 
a bivariant, two; and in a tervariant, one. Nothing is 
gained by taking account at first of all possible com- 
plexities: a very satisfactory idea of the.relationships in 
two-component systems may be obtained from a considera- 
tion to which certain limitations are attached. It is pro- 
posed, therefore, to take as a basis for discussion a general 
ease in which only a single phase may exist in the liquid 
as in the gaseous state, and where the solids may be only 
the single pure components.* To pass from this relatively 
simple case to others where more liquid phases and where 
either crystalline modifications of components or com- 
pounds or mixed crystals may appear, is not a matter of 
ereat difficulty. 


* Subject to the qualification laid down in the Preface, p. 10, 
D2 


52 Trvo-Component Systems 


The 3-dimensional p-T-c model, Graphical representation 
in three dimensions is necessary, and the most suitable 
method is by means of rectangular co-ordinates. Along 
two of them (4C and BD in Figure 11) pressure and tem- 
perature, respectively, are plotted; while along the third, 
AB, composition is given in fractions or percentages of the 
two components. Molar fractions or molar percentages are 
most satisfactory from the more usual points of view, 
though percentages by weight are often preferred. This 
is departing from the representation of specifie volume 


C 


> Pressure —> 





Figure 11. 


adopted in the preceding chapter, but obviously it amounts 
to the same thing in that from a knowledge of the density 
of any phase the amount of each component in unit volume 
may immediately be deduced. 

The point 4 corresponds with pure (or 100 per cent.) 
first component 4; B with pure second component 6. A 
point # would represent a system of which the fraction 
EB/AB would be component A and HA/AB component B. 

This method of plotting has been adopted in Figure 12, — 
which will now be examined in some detail. The figure is 
taken from Bakhuis Roozeboom’s book, Die heterogenen 
Gleachgewichte vom Standpunkte der Phasenlehre, vol. ii 
(Vieweg und Sohn, Braunschweig). It should be kept in 


= 


Two-Component Systems O38 


view throughout practically the whole of this discussion of 
two-component systems. 


-) 
yy 
a 
Y 
x 





Figure 12. 


Section showing phase regions. An isobaric section across 
the upper portion of the model at right angles to the pres- 


54 T'wo-Component Systems 


sure-composition plane will give a preliminary general idea 
of the division of space between the phases. Such a section 
is represented in Figure 13 and shows the progress of 
change from solid (S,+4g) at low temperatures through 
liquid (Z) at intermediate to vapour (/) at high temperatures. 
Between these single-state (though, in the case of the 
solid, not single-phase) regions there come heterogeneous 
areas where solid and 


| i 
Mt | Je || liquid, or liquid and 


| vapour, coexist. A 
—— system, for example, 
=a ? | 





o4 
“I 
~— of the mean composi- 























> tion represented by a 
t = point @ will not exist 
o fl stably asa single phase, 
qb = } but will separate into 
P Seb \ solid and liquid. The. 
B solid will be pure 4, 














and the fraction ac/bc 
S + 5 of the total complex 
will be present as such. 


mre The remainder, ab/be, 
will be a liquid of the 
composition c. Similarly, of a complex represented at d, the 
fraction df/ef will be a liquid of composition e, and the 
fraction ed/e/'a vapour of composition fA 
* At the pressure at which this section has been made, 
the three regions of solid, liquid and gas do not touch at 
any point. Only single-phase or two-phase systems are 
possible. The pressure is taken to be greater than the 
vapour pressure of the solid at any temperature at which 
the latter can exist: or, in other words, the pressure 
selected.is too great to permit vapour to be formed from 


Figure 13. 


“4 


Two-Component Systems Bo 


a solid. But, naturally, at smaller pressures this relation 
may be reversed, so that in the lower part of Figure 12 
a section would show a much less simple division, new 
heterogeneous equilibria becoming: possible. 

The liquid-vapour region O,CDCz. The systems at higher 
pressures will first be considered, beginning at the higher 
temperatures. At sufficiently high temperature, gas alone 
may exist. With fall of temperature liquid appears, and, 
provided the pressure be high enough, gas may disappear. 


D 
I 
w 
E| 
2) 
® 

a 





Composition 


FIGuRE 14. 


The general relations are shown in the portion O,C DO, of 
Figure 12, reproduced by itself in Figure 14. 

The curve 4,C gives the vapour pressures at various _ 
temperatures of the pure liquid component 4: 4,D those 
of the pure component B. For any given temperature the 
points 4,, B,, &e., correspond, and are joined respectively 
by two curves 4,1,B,, 4,V,B,, &c., of which the upper 
gives compositions and pressures of the liquid containing 
both components, and the lower of the vapour in equili- 


‘brium with it. In the solid model formed from an infinite 


56 Two-Component Systenrs 


number of such isotherms there are therefore two surfaces 
which may be called liquid and vapour surfaces, respec- 
tively, joining together along the limiting lines 4,C and 
B,D, and also along the line CD, the significance of which 
will be discussed later. 

Ideal isothermal (p-c) sections of Os,CDOg. According to 
the natures of the eomponents the forms of the isotherms 
A, L, B,V,A,, &e., differ 
considerably. It is easy 
to deduce their courses 
for an ideal system, 
that is, for one in which 
Raoult’s Law connect- 
ing partial pressures 
and molecular concen- 
trations holds for the 
liquid, and in which 
the ideal gas laws hold 
for the vapour. In 
Figure 15, Py, is the 
x Bp Vapour pressure of pure 
liquid component 4 at 
a given temperature, 
and P; that of pure B. Any point ¢ corresponds with 
a composition w of A and 1—a& of &, best expressed in 
this case in gramme-molecules. 

Raoult’s Law may conveniently be put in the form that 
the partial pressure of either constituent in a mixture is 
equal to the product of its vapour pressure when pure and 
its fractional molecular concentration in the mixture. 


Par 


Pressure —>» 





<—1- XK —> = ————_ X_ — 
Figure 15. 


It follows that for a composition ¢ the partial pressure 
of 4 will be #.;, which is equal to ca, where a is the 
intersection of a vertical line from ¢ with the straight line : 


T'wo-Component Systems 57 


P,b. Hence 4,6 gives the partial pressures of 4 in 
mixtures limited at the one end by pure 4 and at the other 
end by pure #. Similarly, 47, gives the partial pres- 
sures of pure B in all these mixtures. For the point ¢, 
the two pressures are ca and cb; the sum is cm, where m 
is on the straight line 2,};. Thus, P,P; is the curve 
(straight line) giving vapour pressures of all liquid mixtures 
from pure A to pure B. 

The compositions of the corresponding vapour mixtures 
follow at once if the simple gas laws be expressed by 
saying that, at a given temperature, the molecular ratio of 
the constituents in the vapour is the ratio of their partial 
pressures. Hence, for a liquid mixture of composition ¢ 
(Figure 15) the vapour in equilibrium will contain 4 to B 
in the proportion ac : dc, or say, #,:1—a, We have then 

a, Oo Le 

I-a,~ be” (1-2) BR (1-2) * B’ 
from which it follows that #, (vapour) will always differ 
from @ (liquid) except in the single case where P, 
happens to equal P;. In that case both liquid and 
saturated vapour will be represented by a straight line 
Pf of constant pressure. In all other cases the com- 
positions of liquid and vapour will differ. If 2, be 
greater than /3, v7, is greater than 2, so that the vapour 
phase contains more than the liquid of the constituent 
with the higher vapour pressure: so, also, if £3 be 
greater than ,. For any particular relation of /, to 
F it is therefore possible at once to calculate the com- 
position of vapour at pressure equilibrium with any liquid 
mixture. The form of the curve is hyperbolic and is 
represented by Pyv/, in Figure 16. ; 

It flattens to a straight line and coincides with P,P; 
when this line is horizontal; but as £, differs more and 





5B i a, Two-Component Systens 


more from Py it becomes further removed, as is shown 
for the cases B, and B,. The vapour curve is, of course, 
symmetrical, but not with respect to the composition axis. 
Hence, the liquid differing most in composition from its 
vapour is not that which contains the components in 
equal molecular proportions, but one containing a greater 
fraction of the less volatile component. This may be seen 
in the lines v/ and v,/, in Figure 16. 
Konowalow’s Law. 
4¥, A general law enun- 
ciated by Kono- 
walow concerning 
the relative posi- 
tions of the liquid 
=p, and vapour pressure- 
~ composition curves 
for binary mixtures 
is to the effect that 
the vapour phase, as 
~P, compared with the 
liquid, always con- 
tains more of that 
SO B component which, 
ate ag on addition to the 
liquid, raises its vapour pressure. Thus, it will be noted 
in Figure 16 that the vapour phase at v, say, at a given 
pressure, contains more of 4 than does the corresponding 
liquid phase /, and that 4 is the component the addition 
of which to Z raises the pressure. 

Usual types of isothermal section of OsCDOz. But since 
Raoult’s Law does not hold except in the limiting cases 
where the proportion of the one constituent or the other 
is small, it follows that this ideal case of Figure 16 is not 









> Pressure — 


Two-Component Systens 59 


found in practice unless the components have marked 
chemical similarity. The liquid curve usually deviates 
considerably from the straight line, lying sometimes below 
and sometimes above it. The simplest case is that repre- 
sented in Figures 12 and 14, and from it, shown again in 
Figure 17, much information of practical value regarding 
the general behaviour of binary liquid mixtures may be 
deduced. Here the 

liquid curve runs 
ews ye rom PN L 
P, to Ly above the 
line 242,; the va- 
pour curve lies be- 
low it. 

Any mixture cor- 
responding with a 
point in the area 
between the curves 
will separate into Py 
two phases, liquid 
and vapour. Above 
the liquid curve, aan 
only liquid exists; A Caq Cc. CewCa B 
the pressure is too ee 
high to permit the 
formation of vapour. Below the vapour curve only vapour 
may exist. 

Lffect of isothermal change of pressure. If now any mix- 
ture, say that of composition and pressure given by /’ in 
Figure 17, be subjected to a steady increase of pressure 
from Py to #,, the alterations which the system undergoes 
may be read at once from the figure. Constancy of tem- 
perature is assumed, so that pressure changes in this and 


- 


Pressure —> 


j= — eee oe -e- - - - - - - - - e 








Figure 17. 


60 Two-Component Systems 


any similar cases must be effected at a rate sufficiently slow 
to permit any necessary addition of heat from, or loss of 
heat to, surroundings to take place. 

Up to, and including, the pressure corresponding with 
the point v, nothing happens to the system upon com- 
pression but change of volume. But an infinitesimal 
increase of pressure above v will result in the production of 
an infinitesimal amount of liquid phase of composition C(a) 
corresponding with the point a on the liquid curve. The 
proportion of liquid will steadily increase with increase of 
pressure, and as its composition differs from that of the 
original vapour, the vapour composition must alter. At 
the point m there will be present liquid corresponding 
with 4 and vapour with d: and in the whole system, which 
may conveniently be termed a ‘phase complex’, the mean 
composition of which is C, there will be present the frac- 
tion md/bd of liquid of composition C(d) and bm/bd of 
vapour of composition C(d). The proportion of vapour 
steadily decreases. Atwit is only xe/ce of the total. At 4, 
on the liquid curve, it has reached zero. The system is 
once more a single phase of composition C, but now liquid. 
Further increase of pressure to Z results merely in com- 
pression of this one phase. Reduction of pressure from L 
to V will bring about precisely the same changes in the 
reverse order. 

Separation of components by isothermal evaporation (or 
condensation). In all this it has been assumed that the 
changes have occurred in a closed system without removal 
at any stage of any of the original mixture. But where 
separation of components is aimed at, as may be the case 
in laboratory or works practice, similar considerations may 
be applied. In. practice, removal of vapour (or liquid) 
progressively formed would be made continuous, but, for 


Two-Component Systems 61 


following the changes graphically, removal at regular 
intervals may be assumed. 

If a liquid mixture of composition C and at pressure LC 
(in Figure 18) be subjected to decreasing: pressure without 
removal of any phase, it will, at pressure ’C, have been 
converted completely to vapour, and the liquid phase will 
have changed from composition / to an infinitely small 
quantity of compo- 7 
sition /,. No liquid 
néarer in composi- 
tion to pure / can 
beobtained. If, how- 
ever, thereduction of 
pressure be stopped 
-at,say,a,midway be- 
tween Land /, then 
liquid and vapour, /, 
and v,, will be pre- 
sent in proportions 
readily calculated. 
If v, be now remov- 
ed and /, subjected 
to a pressure de- 





Composition B 


Fieure 18. 
erease /,4, it in turn 
will separate to 7, and 7. The amount of liquid /, will this 
time not be infinitesimal, but will be the finite fraction 
av, bV 
Ar, 1,7 
of the original /: and it is clear that the more often, and 
hence the smaller the quantities in which, vapour is 
removed during the interval of pressure reduction L/’, the 
greater will be the amount of liquid /, at the end. Then 


/, may be similarly partially vaporized to /, and v,, and 


62 Two-Component Systems 


v, removed: /, to /, and v,, and v, removed: and so on. 

The amount of liquid is steadily decreasing as it approaches 

B in composition, and, of course, its amount is infinitesimal 

when it becomes pure 6. A perfect separation of B in 

finite amount is thus impossible, but a separation of L 

into fractions may be effected, and these fractions will 

differ the more widely in composition : 

(i) the smaller the volume of vapour formed in contact 
with liquid prior to its removal ; 

(ii) the more effectively the diffusion of successively pro- 
duced vapours with one another, while still in the 
space above the liquid, is prevented ; . 

(i11) the more rapidly vapour is removed. 

Quite similar reasoning may be applied to the separation 
of components by fractional condensation of a vaporous 
mixture, with regular removal of liquid formed, leaving 
ultimately vapour of pure A. 

Fractionation by isothermal evaporation or condensation 
of this kind is not so frequently applied in practice as 
isobaric evaporation or condensation. The latter will be 
considered in a similar way presently. 

Other types of isothermal (p-c) sections. Although many 
cases of binary mixtures conform to the type of Figures 
17 and 18, there are numerous others in which the liquid 
and vapour curves may not run steadily downwards or 
upwards from one pure component to the other; that is to 
say, in which the vapour pressures of mixtures may not all 
lie between the values for the respective components. In 
such cases the curves must pass through maximum or 
minimum points and take one or other of the forms shown 
in Figures 19 and 20, 

A maximum is found on the side of the more volatile 
component 4, a minimum on that of the less volatile B. 


Two-Component Systems 63 


To each section PL 
and LF, of the li- 
quid curves in these 
figures must corre- 
spond a vapour sec- 
tion PV and VP, 
respectively. The 
generalization of 


Konowalow, men- — 


tioned above, which 
is based only on an 
assumption that the 
systems to which it 
is applied are in sta- 
ble equilibrium, ‘re- 
quires these to lie 
in the relative posi- 
tions shown. ‘The 
two curves touch at 
the maximum or 
minimum point as 
the case may be, and 
the systems are in 
accord with a gene- 
ral rule deduced by 
Gibbs to the effect 
that at a given tem- 
perature the vapour 
pressure attains a 
maximum or a mini- 
mum whenever the 


two phases become 


alike in composition. 


L LIQUID | 





oo 


Pressure —> 











tL 


as Composition 


Fiaure 19. 





@ 

bes 

3 

“” 

19) 

1-8) 
& > 
A. Composition man 

FIGuRE 20. 


64 Two-Component Systems 


If now there be applied to Figure 19 similar considera- 
tions to those already advanced for Figure 18 regarding 
isothermal distillation of liquid or condensation of vapour, 
with continuous separation of one phase as formed, it may 
readily be seen that for any mixture on the left of the 
maximum point the process will lead in the limit to pure 
liquid 4 and to vapour of the composition of the maximum 
point: on the right 
to pure liquid B and 






VAPOUR the same vapour. A 
separation into A 
and # is thus im- 
possible. Liquid or 


vapour at the maxt1- 
mum point will eva- 





porate or condense 


Temperature —> 


without change of 
composition just as 





S 


a simple substance 
would. 





Similarly in Fig- 





Cam B ure 20 mixtures to 
Sey the left of the mint- 

Bleu er: mum point may be 
fractionated in the limit to pure 4 vapour and liquid of the 
composition of the minimum point: mixtures on the right tio 
pure B vapour and the same liquid. Again, 4 and & can- 
not, by isothermal evaporation or condensation, be obtained 
separately, as in the cases covered by Figure 18. 

Isoharie (T-c) sections of O,CDO,z. This discussion has 
proceeded from a consideration of isothermal sections of 
Figure 14. Two other types of section are possible, viz. 
those at constant pressure and those at constant composi- 


Two-Component Systems 65 


tion. An isobaric section is given in Figure 21, and the 
curves are, in common phraseology, the boiling-point 
curves of binary mixtures. The case usually investigated 
is, of course, that of atmospheric pressure. 

The space corresponding with conditions under which 
liquid alone may exist lies in the lower part of this section, 
and increase of temperature leads first to the heterogeneous 
area 7),/Tpv7',, and 
then to the homoge- 
neous vapour area. 

The behaviour of 
a liquid mixture 
during isobaric dis- 
tillation (or con- 
versely of a vapour 
mixture during’ iso- 
baric condensation) 
is readily predicted 
from the figure. If 
vapour be not re- 
moved as formed, 
the sequence of 
changes is the fol- 
lowing : 

A mixture of temperature and composition represented 
by Z remains homogeneous till the temperature reaches 
that of 7, when an infinitesimal amount of vapour of 
composition @ is formed. With further heating the pro- 
portion of vapour steadily increases. At m the amounts 
of vapour and liquid are as dm : mé, the respective com- 
positions being given by J and d. At x there is still more 
vapour, now of composition ¢ as compared with liquide. Atv 


the liquid has just vanished, its last trace having the composi- 
2568 E 







Temperature —> 


A Composition B 


Figure 22. 


66 T'wo-Component Systems 


tion f, The system is now entirely vaporous and identical in 
composition with the original J. Vapour has thus varied be- 
tween the composition limits a and v: liquid between / and f. 

If, however, the vapour be steadily removed as formed, 
the sequence of changes may be read from Figure 22. 

The liquid Z remains homogeneous until at 2 vapour 
begins to form. If the first removal of vapour, v,, be effected 
when the temperature corresponds with a, a liquid /, remains. 
This on further rise of temperature, 4,0, gives liquid 7, and 
vapour v,; /, at ¢ gives /, and v,; and soon. ‘The amount 
of liquid of composition and temperature corresponding 
with /, is 

av, 60, "Oe 
of the original quantity. The case is very like that of 
isothermal evaporation discussed in connexion with Figure 
18, and similar conditions determine the amount of residual 
liquid obtainable for a given rise of temperature. 

The types of isobaric sections for the cases when the 
isotherms are similar to Figures 19 and 20 are readily 
drawn, and the course of distillation or condensation at fixed 
pressure, with or without removal of one phase as formed, 
does not present anything new. 

Constant composition sections (p-T’) of O, CD Op. There 
remain now the sections of Figure 14 at constant composi- 
tion, one of which is shown by the curve 

DLL SY 
The co-ordinates Heke are temperature and pressure, and 
part of the curve is reproduced in Figure 28 as LS/, 

The distance ZV between the curves on the vertical line 
for any particular temperature gives the difference in pressure 
which results from isothermal vaporization or condensation. 
At pressure /. the mixture is entirely liquid: at P en- 


T'wo-Component Systems 67 


tirely vaporous. At a certain point S the two curves merge 
into one another. This holds for all mixtures, and the series 
of points gives the line CSD of Figure 14 shown as a pro- 
jection in Figure 23, ‘The phenomenon at each point is 
similar to that at the critical point of a simple liquid: the 
end points, C and J, of the line are, in fact, the critical 
points of the individual components 4A and B. Hence 
the name ‘ critical 
curve’ is usually 
given to CSD. 
While it is not 
possible fully to dis- 
cuss this curve here, 
one matter of con- 
trast between the 
points of the criti- 
eal curve of binary 
mixtures and the 
single critical point 
of a pure liquid wv 
should be men- 
tioned. In the lat- Mabaso 
ter case the point LS sak aera 
is at the highest | raced 6 
pressure and the highest temperature at which a two-phase 
system can exist: but with a mixture, the highest pressure 
and highest temperature of coexistence of two phases are 
not necessarily the values of these co-ordinates at the point 
where liquid and vapour phases merge into one another. 
Thus, in Figure 28, P is the point of highest pressure and 
T of temperature, while S is the critical point. 

There are other possibilities, too, and one of them is 
shown in Figure 24. A mixture subjected to the isothermal 

E 2 





Pressure —+ 





68 T'wo-Component Systenrs 


changes indicated by the line abe will pass, in a case like 
this, with increasing pressure from a single vaporous phase 
to a liquid-vapour system, the liquid phase first appearing 
at 6. At ec it will become wholly vaporous again, 
and above that no pressure will liquefy the vapour. 
For fuller discussion of ‘retrograde phenomena’ of 
this kind, the original papers of Duhem (Journ. Phys. 
Chem., 1897, 1, 278 and 
1901, 5, 91), Kuenen 
(Zeit. phys. Chem., 1893, 
11, 38 and 1897, 24, 
667) and van der Waals 
should be consulted. 


Provided that no new 
phases appear, the two- 
surface model of Figure 
14 will presumably ex- 
tend continuously to ab- 
solute zero of temperature 
and to zero vapour pres- 
sure. But the formation 
of a solid phase is the 
usual result of passage to 
lower temperatures, and under conditions represented by 
points on the curves 0,/ and O,F of Figure 12 new 
equilibria enter. Before considering these, however, the 
case represented by the upper left-hand portion of Figure 12, 
shown already in section in the lower part of Figure 18, 
will be discussed. 

Lsobarie sections (T-c) such as XUPVZ of Figure 12, 
Here only solid and liquid phases enter. Pressures are too 
high and temperatures too low for a vapour phase to exist. 


Pressure — 





Temperature —> 


Figure 24, 


Two-Component Systems 69 


The number of investigated cases included in this general 
type is considerable, and from some practical points of view 
it is the most important portion of the two-component 
model. 

As indicated in Figure 12, the effect of pressure on these 
liquid-solid or solid-solid systems is comparatively slight, 
and usually little change in the isobaric sections is to be 
noticed over a con- 
siderable range of 
pressure. In Figure 
25 is given such a 
section, in which t 
temperature and 
composition, respec- 
tively, are repre- 





sented along the t 
axes. The lower 2% 
limit will be abso- 3° 
lute zero. The up- = 
per limits, where & 


vapour appears, 
have already been 
shown in Figure 
13. 

T, is the melting-point of pure 4 under the selected 
pressure: 7, of pure B. ‘The addition of a smali amount 
of the one to the pure melt of the other will lower the 
temperature of equilibrium between solid and liquid. In 
accordance with the laws of dilute solutions, this lowering 
will be proportionate to the number of molecules added, 
and if the compositions along the axis d4 be expressed in 
molecular fractions, the curves for lowering of melting- 
points will be straight lines. As, however, these simple 








A Composition B 


FIGureE 25. 


70 Two-Component Systems 


laws hold only for dilute solutions, the curves from 7, and 
7, usually show an approximation to straightness over only 
the first parts of their courses. 

Along 7,2 are represented the liquids of steadily in- 
creasing concentration in B, which are in equilibrium with 
solid 4,* represented along the vertical line 7,C. Any 
mixture of components, or phase complex, of mean compo- 
sition corresponding with a point, such as J, within the 
area 7,CE will resolve itself into solid 4 (point c) and 
a solution on 7',/, such as a, where a, 4 and c lie on a hori- 
zontal straight line. Of the total complex, ab/ac will be 
solid 4A, and dc/ac will be liquid. 

The number of degrees of freedom possessed by any two- 
component system in two phases is two, but one of these 
having been exercised in the present case in an arbitrary 
selection of pressure, such a system has become univariant. 
Any line like adc may thus be termed a univariant conju- 
gation line, and the area 7,CZ a univariant area. 

Precisely similar considerations may be applied to the 
area 7, DE and its bounding and conjugation lines. 

Hutectic temperature and composition. At H equality in 
concentration is reached in the two series of liquids. Here, 
therefore, solid 4, solid B and solution exist in equilibrium. 
Three phases of two components constitute a univariant 
system, which, under a selected pressure, becomes invariant. 

The invariant point /# is obviously at the lowest tempera- 
ture at which, under a given pressure, a liquid phase can 


* Again the student may be reminded that the solids in equilibrium 
with successive liquids along 7, cannot be identical, that is, perfectly 
pure A, though in the case under consideration differences in composi- 
tion from pure A may be negligibly small. It has already been stated 
in the Preface (p. 10) that for reasons of simplicity in presentation the 
exact compositions of solids will not be taken into account at this 
stage. See also the opening part of Chapter VIII. 


Two-Component Systems 71 


exist, and its composition is such that it can coexist with 
both components. This amounts to saying that the 
particular proportion of solid components corresponding 
with / is one that melts completely at a lower temperature 
than any other. Hence it has become customary to call it 
the ‘eutectic (Gk. evrnxros, easily-melting) mixture’, or 
more briefly, the ‘eutectic’, and the point / the ‘ eutectic 
point’. In the earliest cases investigated, the components 
were, respectively, water and some salt or other, and because 
the eutectic temperatures where ice and solid salt coexisted 
with a certain solution were always below, and often many | 
degrees below, the freezing-point on the thermometric 
scale, the name ‘ cryohydric temperature’ (Gk. xpvos, frost; 
vdwp, water) came into use. There is no particular reason, 
other than that of historical interest, for retaining such a 
term, and it might be as well to abandon it altogether. 
A eutectic point, in all these binary systems, varies slightly, 
though only slightly, with pressure, and measurements are 
customarily made under a total pressure equal to one 
atmosphere. 

Returning to Figure 25, the horizontal line CD, an 
invariant conjugation hne connecting the compositions 
C, H, D of the three coexisting phases, is the lower boun- 
dary of both the liquid-solid univariant areas 7,CH and 
T,DE. Below it only solids can exist. If, then, the 
eutectic solution / be cooled, there will separate from it 
solid A and B side by side. That this must be so will be 
evident if one supposes / supercooled to e, for a solution e 
is supersaturated with 4 and & relatively to the unstable 
prolongations of 7,H to a’ and TH to 4’, respectively. 
The mean composition of the two-phase solid will be the 
same as that of the liquid, so that the freezing-point 
phenomenon recalls that of a single component. Since, 


72 T'wo-Component Systems 


however, the composition of the eutectic varies (however 
slightly) with pressure, the constancy of the freezing-point 
at a given pressure is no more evidence of the presence of 
a single phase than was the identity of composition of 
liquid and vapour during isothermal evaporation at the 
point Zin Figure 19. The proportions of 4 and B in the 
solidified eutectic mixture will obviously be HD/CD and 
EC/CD, respectively. 

Changes with altering temperature. If now any liquid 
mixture, say that of the point xz, be cooled, the changes in 
it may be quantitatively predicted from Figure 25, Until 
the temperature reaches that of x,, on the curve 7,/, cool- 
ing does not bring about any phase reaction. Below 2,, 
however, separation of solid 5 begins, the residual liquid 
becoming therefore relatively richer in 4 and moving along 
the curve #,#. At the temperature of, say, v,, the fraction 
ay/yz of the original mixture will be present as solid B (z), 
and the rest as solution of composition y, the points y, x, 
and z lying on the horizontal conjugation line. At a, on 
CD, the fraction that is liquid, 7,D/EHD, has decreased still 
further, and this liquid is now of eutectic composition. On 
further cooling, the latter will solidify as a whole. These 
proportions then are those of B and eutectic solid, respec- 
tively, in the total solid. Since the eutectic is itself com- 
posed of 4 and B separating side by side, the total com- 
position may also be expressed as Cz,/CD of B and #,.D/CD 
of A. In the process of cooling from a, however, one may 
imagine crystals of B steadily growing, and at «, the eutectic 
mixture filling the interstices between them and then 
quickly freezing to a mixture of comparatively small crystals 
of 4d and B. Hence, there is some reason from a purely 
descriptive point of view for preferring to describe the solid 
in terms of the more sharply contrasted portions B and 


Two-Component Systems 73 


eutectic solid, rather than in terms of 4 and B. In any 
text-book on metallography (for example, Desch, Metallo- 
graphy, Longmans, Green & Co.) will be found photo- 
graphs of magnified polished solid surfaces of two-component 
alloys which illustrate this point. 

Cooling below #, does not bring about any further 
changes in phase. The solid components remain together 
in the proportions formed at w,. The area is thus one of 
the coexistence of two solid phases. 

All such curves as 7,7 and 7,F of Figure 25 lie upon 
the surfaces UO,HP and VOgzEP of Figure 12, or their 
continuations upwards. These surfaces, then, separate the 
space in which liquid alone exists from those spaces in which 
solid may coexist with it. 

Modifications due to formation of compounds. The relatively 
simple case of Figure 25 may become more complicated if 
the two components are able to unite to form one or more 
compounds. Where 4 and & are metals ever so many 
cases of the kind are known: so, too, where one is a solvent 
such as water and the other a salt. Indeed, the case of 
Figure 25, though frequent with metals, is not altogether 
usual with common solvents and solutes. 7',/ is the 
freezing-point curve* of increasingly concentrated solu- 
tions: Z,/ is the solubility curve * of the anhydrous solute. 
It is seldom that one finds a complete curve from the 
eryohydric or eutectic temperature to the melting-point 7’, 
of solute. Figure 26 represents the chief features of the 
more usual systems. 

7',/ is the freezing-point curve of solutions containing 
progressively increasing amounts of component B. At L, 

* It is often convenient to use these familiar terms, though it is per- 


fectly obvious that there is no difference in kind in the two curves, so 
that, from a pedantic point of view, the practice is unjustifiable. 


74 Two-Component Systems 


a eutectic (cryohydric) point, it meets the curve CH, the 
liquids of which are in equilibrium with a solid compound 
of composition corresponding with the point C, say 4,B,. 
At C the compositions of liquid and solid are identical, so 
that this is the melting-point of the compound. A perpen- 
dicular DC’ through C will now cut off a portion 7,4C’CD 
of the whole figure which is in every way similar to 
Figure 25. The two 
components are A 


p 











elf a and a compound 
A: D ! containing this and 
\i i Alp 2B, instead of two 
—\b 7 : f= quite distinct ele- 
d—\ 0 Cc kc ments or  com- 
Al. LEN f= pounds, 4 and £. 
== year eee | Just as the freez- 
i ae ing-point of A,B, is 
S e n= | lowered from C by 





F the addition of 4 
to the melt, so is 
it by the addition 

oy , a aoe of B. Hence, on the 
eae other side of CD a 

Bester curve Cis obtained 

which meets, at, the second eutectic for the system, the curve 

TRF for the liquids in equilibrium with solid 6. The whole 

system is really nothing more than Figure 25 twice over. 

The curve //’ which runs from the one part to the other is 

continuous, that is to say, there is no sudden change of 

direction at the maximum point C. ss 

The division of the whole field into its various systems of 
phases will be readily apparent from the figure. 

Changes with alteration of temperature. It follows from 


Temper 





Two-Component Systems 75 


what has been said that one is always able to read from 
such figures as 25 and 26 the way in which any mixture of 
the components will resolve itself into phases and the pro- 
portions between the phases. Such diagrams give precise, 
quantitative information about any possible binary mixture 
at any temperature, a fact which is not fully appreciated 
when, as is so often the case, attention is directed only to 
curves and their intersections and not to the equally im- 
portant areas of which the curves are merely the boundaries. 
By way of example, the behaviour of any mixture of A 
and £6 in such systems as those of Figure 26 may now 
be deduced. All mixtures represented by points above 
L,HCHT, ave completely liquid. The changes occurring 
on cooling depend upon the particular curve above which 
a point may lie. Any mixture such as that at a, lying 
above 1,H, remains liquid until it cools to the temperature 
of 4 on the curve 7,/. Here an infinitesimal amount of 
solid 4A separates. The amount steadily increases as the 
temperature falls, and at ¢ the fraction ce/ed of the total will 
be present as solid 4 and cd/de as solution of the com- 
position ¢, richer, of course, in B than the original a. 
‘Thermometric arrests. If the cooling agent be constant 
throughout, it will be found that from a to 6 the rate of 
fall of temperature is greater than from 6 downwards. For, 
in accordance with the principle of Le Chatelier and Braun, 
the change in the system, beginning at J, is of a kind to 
counteract cooling; that is to say, it is an exothermic 
change, the heat set free being that’ commonly referred to 
as the latent heat of solidification of the separating com- 
ponent 4. Hence, as compared with a, dc shows a ‘ ther~ 
mometric arrest’ of cooling which continues until the point 
f is reached. Here, f#/Hg has solidified, and the rest of 
the original mixture, the fraction /g/Hg, is the eutectic 


76 T'wo-Component Systems 


liquid of composition #, Any attempt to cool this liquid 
further results in its solidifying to a eutectic mixture, 
similar to that of # in Figure 25. In this case, however, 
the two phases of the eutectic are solid 4 and the solid 
compound 4,8, of composition C; Hh/gh of the liquid 
going to the former, and Ly/gh to the latter. The total 
solid obtained may be represented in proportions of the ~ 
original weight taken either as fg/gH of eutectic of the 
composition just quoted and fH/7F of solid A, or else as 
fi/gh of solid A and /g/gh of solid compound 4, B,. 

Since the eutectic liquid # solidifies as a whole, the 
phenomenon is different from that of the passage along 
the line 0/, where there is a continuous adjustment of com- 
position of the liquid phase. The case is, in fact, quite like 
that of the freezing-point of a pure substance. Any attempt 
to cool the mixture results, by separation of solid, in pro- 
duction of heat sufficient to annul the cooling and keep the 
temperature at that of # until all liquid has solidified. 
Hence, there is here a much more marked thermometric 
arrest than along Jf under constant cooling agency. 

In the many difficult investigations which have been 
carried out upon binary mixture of metals, rates of cooling 
(‘thermal analysis’) have been of great importance in 
determining equilibria. It is not possible here to go 
further into a practical question of this kind. Details will 
be found in Desch’s Metallography (Longmans, Green & 
Co., London) or Giua’s Chemical Combination among Metals 
(J. & A. Churchill). 

The cooling of a mixture a’ above FT), is, of course very 
similar to that of a A mixture & above C¥ (or, mutatis 
mutandis, k’ above HC) will remain liquid to Z when the 
solid compound 4, £8, will begin to separate. At m the 
fraction om/on will have solidified and mn/ou will still be 


Two-Component Systems 77 


liquid though now of composition 0. At yp the liquid will 
be the eutectic mixture /, and, on attempted further cool- 
ing, it will solidify without further change of composition 
to the two phases, compound and pure &. The proportions 
in the solid are, as before, at once calculable. 

Region of solid phases. For the regions in which only 
solid phases exist, that is to say, below the boundary 
glithg Fw, the following summary holds: 

In the area AghC’, A and A,B, exist side by side: in 
AgHK’ the mixture may be regarded as one of A and eu- 
tectic of composition £’, and in EH’ ZAC’ as one of 4, B, and 
eutectic. In C’qwB, similarly, 6 and A,B, exist together : 
in C’gFF’ the solid may be considered as a mixture of A, B, 
and eutectic #”, and in /’/wB of eutectic and solid B. 

Changes with alteration of composition, It is of interest, 
also, to make use of a diagram such as Figure 26 for 
predicting the changes which must oceur when, at constant 
temperature, a system undergoes a steady alteration of 
composition ; when, for example, pure 4 at a temperature 
corresponding with 7 receives continuous additions of 
B (or what would come to the same thing, when some 
mixture of 4 and #& steadily loses 4A by, say, evaporation), 
passing through all possible compounds until the limit, 
pure J, is reached. 

_At first the pure 4 is solid, being the interval 77 below 
its melting-point. The addition of B causes partial lique- 
faction, the new liquid phase having the composition of s. 
The proportion of the total mixture which is liquid steadily 
increases until complex and s have the same composition, at 
which point 4 has vanished. Further additions of B 
dissolve; a liquid phase of changing composition from s to 
¢ alone being present. At ¢, however, solid again appears, 
this time the compound 4, B,. The amount of it increases 


78 Two-Component Systems 


with more B, the liquid remaining at ¢ and decreasing in 
quantity until at # this phase disappears and there is 
present only pure solid compound. With more 8, liquid Z 
is formed and increases in quantity until the whole complex 
consists of it, 4,.B, having disappeared. From / to w is 
an interval of liquid solutions of increasing content in J, 
but at w, solid phase, the pure component JS separates. 
Thence onwards addition of solid B has no effect; the solid 
simply accumulates. If, however, the relative amount of B 
be increased by withdrawal of 4, B will continuously form 
and the liquid « diminish in amount to zero. 

The alteration of composition has therefore resulted in 
passages from one solid phase 4 to a two-phase system of 
A and liquid, then to a one-phase system of liquid, to 
two-phase solid 4,8, and a second liquid, one-phase solid 
A,,.B,, two-phase solid 4, 6, and another liquid, one-phase 
liquid, two-phase solid 6 and liquid and finally a single- 
phase system of pure J. 

While all this deduction is quite logical, it is necessary 
to point out that from the practical point of .view the 
realization is not necessarily easy. Thus the addition of 
solid GB at temperature 7 to solid 4 at the same temperature 
will not invariably result in the immediate formation of 
the stable system solid 4 and liquid s. The rate of progress 
to equilibrium may be indefinitely slow, and only to be 
effected practically by passing first through other con- 
ditions: but the final equilibrium states must in all cases 
be those indicated. 

Metastable melting-points of compounds. Only one other 
point remains to be mentioned in connexion with systems 
of this type. In Figure 26 the curve ZF (solid B) meets 
the curve HCF (solid 4,B,) beyond the maximum C. It is 
frequently found, however, that the equilibrium point 


Two-Component Systems 79 


between 4,6, and solid B is reached on the part HC 
below C; that is to say, the liquid with which the two 
solids may coexist is richer in 4 than the compound 4, B,. 
Figure 27 shows such a case. 

C is now a metastable or labile point. If pure solid 4, B, 
be raised to the temperature C, it certainly forms solid and 
liquid, but when the new equilibrium is attained, the former 
(2C/be of the total) 
is pure 5, not 
A, By, and the latter 
(Ce/be of the total 
is of composition 4. ‘Tas ‘| 
Pure solid 4,6, can 3 
thus exist «stably 
only in the presence 
of a liquid contain- 
ine more dA. than 
it itself does; the 
point /’ is the poor- 
est in A of such 


liquids. 

As compared with | 
the interpretation Composition B 
of Figure 26, that PGE Ena ls 
of Figure 27 offers little else that is new. A consideration 
of the changes occurring on cooling a liquid mixture at a 
will suffice to emphasize what differences there are. The 
single phase, liquid, persists until the temperature is that 
of 4. Below this pure B separates, the liquid becoming 
relatively richer in 4 until it reaches the composition of / 
when ci’/fF of the original mixture has separated as B. 
At Fa new solid phase, the compound 4,B,, appears, and _ 
any further attempt to cool the mixture will result in the 


ip 


NS 








SS ex 
€ 
(ar) 


| 














he 


Temperature —> re 


> 


80 Two-Component Systems 


separated B redissolving, while more 4,8, forms. So long 
as any solid B remains, the temperature cannot be changed. 
Since the attempt to cool the mixture results in solution of 
B and the separation of 4, B,, it follows from Le Chatelier's 
theorem that the heat developed in the latter process must 
exceed the cooling consequent upon the former. 

When all & has returned to solution, Yc/Pg of the 
system is present as solid 4, B,, and cg/Fy as solution F. 
On further cooling, more compound separates and the 
solution ultimately reaches the eutectic point # at which 
A appears. ‘The case thereafter is identical with that pre- 
viously outlined in connexion with Figure 26. 

One may express the differences between the cases of 
Figures 26 and 27 in another way by comparing the corre- 
sponding conjugation lines, gw and Ff, respectively. 
The relation between the phases whose compositions lie on 
the former is given by the equation 

Solid 4, B, (q) + Solid B (w) = Liquid (F). 
Between those lying alone the latter it is 
Solid 4,5, = Liquid (F) + Solid B(/). 
The direction of reaction depends upon the direction of 
temperature alteration, the upper arrow in each case giving 
the change occurring with rise of temperature. 

Transition points, Since at the point / in Figure 27 
the change consists in the conversion of, or transition from, 
one solid to another, such a point is commonly termed | 
a transition point to distinguish it from the eutectic point 
of Figure 26. There is one outstanding difference between 
the two which is implicit in the equations above. Increase 
of temperature at the transition point / gives liquid in 
equilibrium with solid B, the solid 4,8, having disap- 


peared: decrease gives liquid with solid 4,B,, solid B 


Two-Component Systems 81 


having diminished to zero. Thus liquids may exist above 
and below a transition point. On the other hand, increase 
of temperature at a eutectic point gives liquid with either 
solid B or solid 4,B,: decrease gives solids only. The 
eutectic point, then, differs from the transition point in 


being the lowest temperature at which a liquid can exist. 


1600 


fo) 


Temperature (°C) 





Pd Atomic Percentages Pb 


Figure 28. 


Some actual ewamples. If now Figures 26 and 27 be 
understood, there will be no difficulty in interpreting such 
diagrams as are given in Figures 28 and 29. The former 
(taken from Giua, Chemical Combination among Metals, 
p. 234), for the binary system lead-palladium, is an excel- 
lent example of the relations found in investigations on 
alloys. The part marked M.C. is rather different from 


256 F 


82 Two-Component Systems 


any case so far discussed and its type will be referred to 
again in Chapter VIII. 

Figure 29 is from the work of Roozeboom (Zeit. phys. 
Chem., 1892, 10, 477) on ferric chloride and water, and is 
typical of so-called ‘solubility work’ on binary systems of 


salt and solvent. These two cases are rather more compli- 


cated than the majority of their kind. 








40 —_— 
: | : 
be 4 
= oe 
i Pay. 2== 
5 ‘| = 
EE = 
o _——— 
FE Ss 
-2 [ ae 
—— 
= 
a —— —— 
ae 10 20 30 40 50 
H20 Molecular Percentages FeCl, 


Figure 29. 


The lower portion of Kigure 12. There remains now for 
detailed consideration that portion of the typical space 
model which is shown in the lower part of Figure 12. 
Here pressures and temperatures may both be low and, in 
consequence, rather more complicated equilibria may occur. 
The most satisfactory way of grasping the significance of 
the model is to study it in successive isothermal sections. 
Much assistance in understanding its various divisions in 


Two-Component Systems 83 


these lower parts will be obtained from the perspective 
sketches shown in Figures 30 and 31, which are taken in 


C 





Ficure 31. 


outline directly from Roozeboom’s Le heterogenen Gleich- 
gewichte, vol. 11, and which should be examined in conjunc- 
tion with Figure 12. 

F 2 


84 Two-Component Systems 


Successive isothermal (p-c) sections. Figure 32 gives a 
succession of isothermal sections illustrating the changes 
which occur when solid phases begin to appear as one 
passes to lower temperatures and pressures in the continua- 
tion of that portion of the model which is sketched in 
Figure 14. They are sections extending from Y to X in 
Figure 30. 

It will be assumed as before that, at a given temperature 
liquid component A has a higher vapour pressure than 


Temp. decreasing —> 



























































Ts 7 Ty Tg T7 
i ee “an 
Nn = K" aa FO 2 a 
“Aa iP nN ‘ T 
SS WAHNTT 
AW U a N tl 
NUP: i 
B BA l, BK 
Composition 


FIGuRE 82, 


liquid 6. Asa rule, though not invariably, it follows that 
B will solidify before 4, and it will be supposed here that 
it does so. The general case is not altered if the reverse 
holds. No compounds between, or mixed crystals contain- 
ing, the components will be taken into consideration. The 
principles involved in these cases are identical, but there is 
of course added complication in the graphical representa- 
tion, and it is not considered worth while to undertake 
their detailed discussion. 
Section 7, of Figure 32 shows the bivariant system 
liquid-vapour, for a selected temperature, already discussed 
in connexion with Figure 17. A, is a point on the vapour- 


T'wo-Component Systems 85 


pressure curve CO, (Figure 12) of pure liquid 4; 8, on 
DOz of pure liquid B, A,/B, gives compositions of liquid 
phases, and 4,vJ/, of corresponding vapour phases. 

As temperature falls, so on the whole will vapour pres- 
sure, but no attempt has been made to represent relative 
pressures in corresponding portions of the various sections 
in Figure 382. At a temperature O, (Figure 12) it becomes 
possible for solid B to separate from its own melt. When, 
however, any 4 is present in the liquid, solid B cannot 
_remain. The partial vapour pressure of liquid B is lowered 
by dissolved 4, so that solid B in contact with a solution 
would inevitably dissolve. But if the temperature be 
lowered it becomes possible for B to exist with solution 
containing A: the case is illustrated by section 7’, of 
Figure 32. B, is here the vapour pressure of metastable 
liquid 6. ‘The pressure of the stable solid is therefore 
lower, say that of the point K. In the liquid and vapour 
phase equilibrium, now, the partial pressure of B steadily 
decreases from 6, with addition of 4, though of course the 
total pressure increases. It is therefore possible to find a 
point at which liquid # and vapour F are in equilibrium at 
a total pressure to which B contributes a value equal to 
BK: or, in other words, there is here a point at which 
solid B is in equilibrium with vapour and liquid both con- 
taining 4 and 4, The portions 7B, and /BA, of the liquid 
and vapour curves are no longer stable systems in presence 
of solid B, since the partial pressure of the latter in them 
is always greater than BK. The curve /K gives the con- 
centration of the vapour phase in equilibrium with solid B, 
and, so long as the proportion of 6 is greater than corre- 
sponds with /, no liquid phase can be formed. 

At a still lower temperature, section 7’,, the phenomena 
remain of similar kind, but the difference KB, is increased, 


86 Two-Component Systems 


and f’and # are further up their respective curves. The 
remaining sections show the steady increase of the region 
FKH of coexistence of solid and vapour, without liquid. 
At some temperature, a little below that of section 7,, the 
melting-point O, (Figure 12) of pure 4 will be reached. 
In section 7',, 4K’ is the vapour pressure of solid 4, and 
AA, that of its metastable liquid. As addition of B will 
lower the latter, a value corresponding with H’ on the 
liquid curve and Ff” on the vapour curve must ultimately 
be reached at which the partial pressure of B in the mix- 
ture is equal to 4K’. We have thus much the same case 
as with B, except that the total vapour pressure at which 
solid, liquid and vapour coexist clearly falls between that 
of pure solid, 4K’, and metastable pure liquid, 44,. In the 
other case it was above both. 

Sections 7, and 7, show further advances in these 
changes. The regions in which solid 4 and vapour, and 
solid B and vapour, coexist are steadily increasing. That 
in which liquid and vapour coexist, without solid, is 
gradually decreasing. Section 7, shows the temperature 
at which the latter just disappears. J and J” coincide: so 
do Hand #’. At the pressure H’/’H’H and temperature T,, 
solid A (H’), solid B (ZH), liquid (Z’) and vapour (£”) exist 
together. It is the one four-phase or invariant system 
possible with two components which show only the relations 
assumed in Figure 12. 

If now these and all intermediate sections be put to- 
gether, they will be seen to form the portion of Figure 12 
running from a little above Og, (and hence Q,) so far 
as the temperature corresponding with the line GYFH, but 
bounded above by the surfaces O, 4G and O,4H. It is 
better visualized from Figure 30 as the portion cut off 
between the levels of the planes Y and X. The appearance 


T'wo-Component Systems 87 


from below of the under portion of Figure 30, the line of 

vision being parallel to the pressure axis, is diagrammati- 
cally represented in Figure 33. The directions of rise of the 
three curved surfaces seen are indicated by arrows, A line 
at or about the level KK’ would correspond with the 
isotherm 7', of Figure 82. 

The close examination and com parison of Figures 12, 
30, 31 and 33 at 
this stage will am- 
ply repay the stu- 
dent. Sufficient de- 
scription has been 
given to enable any 
one to tell what 
phases exist at par- 
ticular points, on 
particular lines and 
surfaces, and in par- 
ticular volumes, and 
hence to appreciate 
the precise types of 
system represented. 

More complete iso- 
therms. Tocomplete — 
the whole discussion and bring more into line its different 
divisions, a few rather more complete isotherms will now 
be given, embracing the region covered by the isobars of the 
type of Figure 25. The form of these will depend upon the 
way in which pressure affects the temperatures of equilibrium 
in systemscontaining solid and liquid bounded by the melting- 
points of the components on either hand and the eutectic 
in between. ‘This comes to the same thing as saying that 
the form of the isotherms will depend upon the slopes of 





B Composition 


Figure 33. 


88 Two-Component Systems 


the curves 0,U, O3V and EP, respectively, in Figure 12. 
The slopes depend, of course, upon the relative specific 
volumes of liquid and solid phases of components or 
eutectic, as the case may be. If, as in the case of water, 
fusion occurs with contraction, then with increase of pres- 
sure the curves will slope towards lower temperatures. It 
will be assumed, however, that the more general case of 
expansion on fusion 
Z holds both for indi- 
vidual components 
and eutectic, so that 
the curves 0O,U, 
OzV and EP will 
be assumed to pass 
towards higher tem- 
peratures as pres- 
sure increases. 
Isothermal section 


wo 











f “Fa just above Op. Fig- 
g ure 34 gives an 
‘ isothermal section 
A Composition B taken a little above 

FIauRE 34. the temperature Op. 


_ The sections should 
be studied along with the models, Figures 12, 30 
and 3]. 

At a sufficiently high pressure, P, liquid can be com- 
pressed to solid. The higher the pressure the more solid 
is formed. As the solid is pure B, the residual liquid 
becomes relatively richer in 4 and hence less readily 
separates B. A succession of equilibria is brought about. 
With a mixture of composition and pressure corresponding 
with z, for example, wy/yz of it solidities to B: xz/yz forms 


Two-Component Systems 89 


liquid of the composi- 
tion y. The rest of the 
figure offers nothing 
new. 

At lower tempera- 
tures P and JF; ap- 
proach and at Op (the 
melting-point under 
ordinary pressure) they 
coincide. 

Section between Oy, 
and Og. At a tempera- 
ture between QO, and 
O; the vertical sec- 
tion is that shown in 
Figure*35. 

The notation in the 
figure sufficiently ex-— 
plains the division and 
limits of the different 
phases. There may 
here appear on the 4 
side of the diagram an 
area S, + L (notshown) 
similar to S,+Z in 
Figure 34. 

Section below Oy. 
Below the tempera- 
ture O,, solid 4 can 
exist, and between O, 
and the eutectic tem- 
perature the section is 
of the form of Figure36. 


Pressure —> 





A Composition B 


FiaurE 35. 











Sa ae L = Sp + h 
(PES ak Sac eT 
[a Soo a pi aa 
Es YS 
-—_———— Pots Viena chats ae eee ae | 
ise Lees [ag f pera ee cars 
Neti NS a ee 
{|S \L+V¥\ ES 
2 y aN 
: U 
: 
VAPOUR Space 
A Composition B 


FIgvRE 36. 


90 Two-Component Systems 


Still nearer the eutectic temperature, it may be that the 
liquid region becomes totally enclosed and the section 
changes to that of Figure 37, cutting through the line /P 
at a point p. 

At all pressures above p, mixtures of A and JS are entirely 
solid, being two-phase and consisting of crystals of the 


y 
Danse 
Sai + Sp 








| See eee 

| eee ae 

|. 

| 2. Se i ae 

Le ee 

a Ve Se 

EN |. 2222 ees 

i Wii ee 
8) 
5 
% 
= 
oo 

A : B 

Composition 
FiguReE 37. 


components side by side as already discussed in connexion 
with Figure 25. | 

With decrease of temperature, the regions denoted by 
S,+L, SgtL, L+V and L steadily diminish until at or 
below the invariant point # the type of isothermal section 
is that of Figure 38, the interpretation of which requires 
no further statement. 

These figures, 34-38, will be seen to include the sections 


Two-Component Systems 91 


in Figure 32, but to show in addition those regions in 
which the vapour phase does not appear. 

If the slopes of the curves 0,U, Op’ and EP be other 
than as assumed, the consequential modifications in these 
diagrams are readily made. It will serve as a good exer- 
cise for a student to sketch the isotherms for such cases 
and also to work out the changes which occur with changing 


lh 


Set On 









































A Composition B 


Figure 38, 
pressure in a system of, say, composition # at the tempera- 
ture of Figure 37. 

There are, of course, numerous points connected with 
two-component systems which have been omitted from 
this discussion, which has, however, traversed the chief 
characteristics. 'The whole subject has been worked out 
very fully by Roozeboom and Schreinemakers, and any one 
wishing for further detail will be well advised to study 
the original works of these distinguished investigators 
and their pupils, 


CHAPTER V 


THREE-COMPONENT SYSTEMS 


General. With the addition of a third component, the 
sum of the degrees of freedom possessed by any system and 
the number of phases present in it becomes equal to five. 
The greatest number of phases that may be present in~ 
general equilibrium is therefore five, being one more than 
in the corresponding invariant system of two components. 

Only one gaseous phase is possible: the remainder may 
be all solid or all liquid (though no actual case is known 
where all are liquid), or some may be solid and the rest 
liquid. Univariant systems contain four phases; bi- 
variant, three ; tervariant, two; quadrivariant, one. 

The maximum number of degrees of freedom in the sim- 
plest possible case of a single phase is thus four; that is to 
say, temperature, pressure and the concentrations of two of 
the components may be arbitrarily varied within certain 
limits. | 

These three-component systems admit of so many possi- 
bilities of coexistence of phases that it is out of the ques- 
tion to deal with them here so fully as was possible with 
two-component systems. It is proposed, therefore, deli- 
berately to omit any consideration of vapour phases, either 
as regards composition or pressure. 

This means loss from the theoretical aspect; but it has 
justification, not only on grounds of brevity and simplicity, 
but also from the practical point of view. For in the first 
place not much work has yet been done on vapour phases in 
ternary systems, and in the second place, the equilibria 
between solids and liquids, which have most usually been 


Three-Component Systems 93 


studied, are so little affected by pressure that one may 
generally neglect the error resulting from failure to allow 
for pressure changes ; or, what comes to the same thing, 
from assuming pressure to be constant throughout a series 
of investigations and of such a magnitude that a vapour 
phase does not form. Of course there are cases where such 
an assumption is not justified ; where, for example, an 
investigation extends over an enormous range of tempera- 
tures. But even after setting these cases entirely aside 
there remains quite a sufficiently extensive field to admit of 
a student obtaining a very sound introduction to systems 
of three components. Moreover, there is the very practical 
point to be considered that in a graphical method of treat- 
ment such as that being followed, the number of variables 
which may be simply represented in a three-dimensional 
figure is limited. Concentrations of systems may no longer 
be plotted, as they were in the preceding chapter, along a 
single axis. ‘Two axes (or a plane surface) are required, so 
that only one dimension remains for the specification of a 
third variable, either temperature or pressure. The influ- 
ence of the latter is so much less than that of the former 
that a selection of temperature as the more suitable third 
variable for representation of general cases is inevitable. 
Systems in which the pressure is taken to be so high 
that vapour cannot form are always spoken of as ‘ con- 
densed systems’. ‘The selection of pressure constitutes the 
exercise ‘of one degree of freedom, so that in such cases four 
coexisting phases will constitute an invariant system, three 
a univariant, and so on. In practical investigations one 
usually does not trouble to maintain pressure at a particular 
value. Equilibria between solids and liquids alter so very 
slightly with moderate or small pressure changes that the 
~ effect of failure to maintain the pressure at a chosen value 


94 Three-Component Systems 


is generally well within the limit of experimental error in 
analysis. 

Graphical methods. Various methods of plotting the 
relations between the proportions of three components and 
the temperature have been, and are, used. In the earlier 
work of Roozeboom and Schreinemakers, a preference was 
shown for the method set out in Figure 39. 

Temperature is here measured along the vertical axis, 
while as ordinates 
and abscissae are 
taken either the 
amounts of A and 
B, _ respectively, 
present for every 
selected unit of C 
(for example, 100 
grammes or a kilo- 
oram-molecule) or 
the proportions 
(say percentages) 
Oo of A and B ina 
Mols.@™9B:1000 Mols.@ms) C. civenteoml ott a 


eee “mixture of 4, B 
and C. Either method serves many purposes, but the 
former suffers, in particular, from a grave disadvantage in 
that pure 4 and pure B cannot be represented in a finite 
diagram, since they are given by points infinitely distant 
from the point of origin 0 (pure C). Both have the dis- 
advantage that the proportion of the third component 
cannot be read immediately from the figure. 

The equilateral triangular diagram. The method sug- 
gested originally by Gibbs is free from these defects, and 
there is no reason at all why it should not be universally — 






Temperature — 


Three-Component Systems 95 


adopted, except just in particular cases where for some 
special reason other methods give clearer representations. 
It will be used almost exclusively in the sequel. 

Gibbs made use of the geometrical fact that from any 
point in an equilateral triangle the lines drawn parallel to 
the respective sides, added together, equal in length each 
side of the triangle. If then each side be taken as of unit 
length, the composi- 
tion of any complex 
may be represented 
by a point so- chosen 
that the parallels are 
the same fractions of 
the unit length as 
they are of the total 
quantity of complex. 
For cenvenience, the 
sides are taken to 
contain 100 units, 
and the proportions 
plotted are then per- 
centages either by weight or by molecule. 

Figure 40 shows precisely how this is applied. 

ABC is an equilateral triangle of side equal to 100 
units. If those parallels which pass from any point D 
within it to the side opposite the angle 4, namely, DJ or 
DK, be taken as the percentage a of component 4; those 
to the sides opposite 6 and C' as the corresponding values, 
6 and ec, of components B and C, respectively ; then it is 
readily seen from a glance at the various equilateral tri- 
angles and parallelograms in the figure that a+0+c equals 
each of the sides 44, BC and CA, or 100 units. Further, 
if D lie on one of these sides instead of within the diagram, 





Fieure 40. 


96 Three-Component Systems 


one of the parallels vanishes and a binary mixture is repre- 
sented. Thus the point fis that fora mixture containing 
BF per cent. of A and Af of B. Points on BC represent 
binary mixtures of B and C, and on AC of A and C. Lastly, 
if D be coincident with A, Bor C, the corners of the tri- 
angle, two parallels vanish. It follows that the corners 
represent the pure single component 4, B or C, as the case 
may be. Temperature may then be plotted at right angles 
to the plane of the triangle. 

All possible mixtures may thus be shown in this tri- 
angular diagram, and the proportions of components corre- 
sponding with particular points may be read immediately, 

The simple principle that if a mixture XY is composed of, 
or can be resolved into, two other mixtures, or components, 
Y and Z, the points representing the three compositions, 
X, Y and Z, must lie on a straight line, which is termed a 
conjugation line, naturally holds in ternary, precisely as in 
binary, systems. Thus a mixture represented by D in 
Figure 40 may be regarded as composed of those given by G 
and J, respectively, in the proportions DJ/GJ of the former 
to DG/GJ of the latter. Or, again, it may be regarded as com- 
posed of pure component # and the mixture Z in the propor- 
tion LD/LB: DB/LB. Moreover, L in turn is seen to have 
a composition which may be expressed additively in terms of 
Aand Cin the ratio CL/CA: LA/CA. There will be abundant 
occasion in the sequel for applying this very simple relation. 

In the case of two-component systems, by omission of 
all complications, Roozeboom developed a comparatively 
simple model (Figure 12) which illustrated the main possi- 
bilities of the case. With three components, even confin- 
ing attention to concentrations and temperature, this 
cannot be done so satisfactorily with a single figure, and 
therefore a succession of models will be employed to show 


Three-Conuvponent Systems a7 


the most commonly occurring relations. A beginning will 
be made with systems in which solid and liquid phases are 
present in equilibria of the simplest types. 

Simplest equilibria between solids and liquids, In 
Figure 41, which aims at showing a three-dimensional 
concentration-temperature diagram in perspective, A, is the 
melting-point of pure 
component 4, B, of B 
and C, of C. The tri- 
angular prism is cut A 
into two portions by the 
three surfaces A,plr, 
Bygkp and  C,rkg, 
which meet in pairs 
along pl, gf and r#. 

Any system repre- 
sented by a point in the 
upper part is entirely 
liquid, This is a single- AR c 
phase region. Below 
the surfaces a number 
of polyphase regions 
exist, each with per- 
fectly definite limits and 3 
all fitting together to make up the lower part of the prism. 

The dissection of the model offers little difficulty. Start- 
ing from 4, where liquid and solid pure 4 coexist, addition 
of B will in general give a solution of it in A which will be 
in equilibrium with solid 4 at a temperature lower than 4, 
in proportion to the amount added. At py no more # will 
dissolve in the melt, but solid 4, B and solution will co- 
exist. This, therefore, is the binary eutectic of Figure 25 
(Chapter IV), 4, being joined here by the similar curve 
G 









Temperature—> 








Fieure 41, 


2668 


98 Three-Component Systems 


Bp. A,pB, lies, of course, entirely in the vertical plane 
above 4B, where the ternary system merges into one of its 
three constituent binary systems. 

If now to the eutectic p in equilibrium with solid 4 
and solid 6 there be added some of the third component 
C, the same phenomena may be expected as are found in 
the two-component case. The added C will dissolve and 
this solution will remain in equilibrium with solid 4 and 
5, but at a lower temperature than that of py. Therefore 
the curve showing the composition of the melt progressively 
alters in a direction downwards. A point # will ultimately 
be reached at which no more C will dissolve. Here, then, 
three solids and a liquid are in equilibrium. It may be 
termed a ternary eutectic point and, there being four phases 
present, it is a (condensed) invariant system. Strictly, 
speaking, four phases constitute a univariant system, but 
one degree of freedom has been exercised in this case in a 
selection of pressure. If, therefore, other pressures be 
exerted upon this system of phases, they may coexist at 
certain other temperatures than that of #2 in Figure 41. 
The effect of pressure, however, is very small indeed. 

Just the same argument may now be applied in explana- 
tion of the sets of curves B, 9, C,q and gH on the one hand, 
and 4,7, C,r and 7/ on the other. One must bear in mind 
that these curves are merely the boundaries of the three 
surfaces which together make up the lower limits of that 
portion of the prism, the (condensed) tervariant system, 
in which the single unsaturated liquid phase exists. Once 
a liquid mixture of composition 2 has fallen from a tem- 
perature at the level to one at the level y lying on one of 
these surfaces the liquid has reached the saturation limit and 
further cooling results in the separation of one or more com- 
ponents as solids. 4,pHr, B,pHg and C,i£g may therefore 


Three-Component Systems oY 


be termed ‘saturation surfaces’, the solid phases with which 
liquids lying on them are saturated being 4, B and GC, 
respectively. Should the liquid composition fall on a line 
pt, gE or rE, common to two surfaces, it is simultaneously 
saturated with respect to two of these components, either 
A and &, B and C or A and C as the case may be. When 
of the composition #, it is saturated with respect to all 
three. 

Continued cooling to points below these three saturation 
surfaces will result, then, in phase reactions taking place 
with production of complexes of liquid in equilibrium with 
one, two or three solids, yielding systems ranging there- 
fore, under a fixed pressure, from bivariance to invariance. 

There are three regions such that any mixture repre- 
sented by a point within them separates into a bivariant 
system of solid and liquid. ‘The first, in which the solid is 
A and the liquid may lie anywhere on the surface 4, pi, 
is bounded by that surface and such others as will be traced 
out by a straight rod of adjustable length moving horizon- 
tally with one end on the vertical lime 44, and the other 
passing continuously from 4, to 7, thence to /, to p and 
back again to 4,. The second, in which the solid is JS, is 
bounded by the surface b,p/q and the trace of a similar 
horizontal rod or line connecting some part of 64, with 
successive points from £, to 7, thence through # to p and 
back to B,. In the third, the solid is C and the bounding 
surfaces may be described similarly. 

There are also three univariant regions or spaces, any 
complex within which separates into liquid and two solids. 
The first, where the solids are A and & and the liquid is 
somewhere along p/, is best described as that marked off 
by a series -of horizontal triangles with apices passing 
successively from p to 4H, and bases of length AB. The 

i Gs e 


100 Three-Component Systems 


second and third may be described similarly with respect: 
to gH and BC in the one case, and r# and AC in the 
other. 

Lastly, there is an invariant system consisting of the 
horizontal plane passing through £. At the temperature 
of this plane any ternary complex will give liquid / and 
the three solids, 4, B and C. 

As to the region below the level of /, so far as ordinary 
experiment will show, 4, B and C may coexist in any pro- 
portions. As there are three phases, the region is, strictly 
speaking, univariant when pressure is fixed. This means 
that compositions may vary with temperature, but for the 
present, as already admitted, strict analysis of the question 
of compositions of solid phases is being avoided. It is a 
useful exercise for a student to make perspective drawings 
of all these sections of the prism. 

Summarizing, we have the following (condensed) 
systems : 

One tervariant : liquid. 

Three bivariant: one liquid and one solid. 

Four univariant : three of them with one liquid and two 
solids, the fourth with three solids. 

One invariant : one liquid and three solids. 

Altogether, the ternary prism has been divided into 
eight distinct portions belonging, respectively, to the eight 
systems first mentioned. The ninth (invariant) system is 
a plane. 

Isothermal sections of Figure 41, A series of {isobaric) 
isothermal sections of general type may immediately be 
deduced from Figure 41. At temperatures above C; all 
possible proportions of 4, 4 and C fuse to homogeneous 
liquids, Figure 42 shows the system at a temperature 
below C, but above 4,. 


» 


Three-Component Systems 101 


All complexes of composition falling within the area 
ABba are liquid ; those falling in a/C separate into two 
phases, of which one is solid C and the other a liquid along 
the curve ad. ‘Thus, the complex c gives solid C and solu- 
tion of composition d, the proportion of solid being cd//dC 
of the total, and of liquid eC/MC. 

The area abC is obviously a portion of one of the bivariant 
spaces described above. In the isothermal section, however, 
one degree of freedom 
has been exercised in 
the selection of tem- 
perature, so that alC 
isaunivariant system, 
or to be quite de- / 
finite, an isothermal, / 
condensed univariant 
system. One must a 
continually bear in 
mind the fact that 
the variance or resi- 
dual variance of a 
specified system de- 
pends not only upon the number of phases present, but also 
upon the number of degrees of freedom which have already 
been exercised in defining the particular conditions under 
which the system is being examined. 

The shape of any saturation curve such as a/, and hence 
of the surface of which it is a part, cannot be predicted. 
It depends, iter alia, upon the structure of the liquid, that 
is to say, upon the presence of, and the proportions between, 
simple or compound molecules, or ions, or whatever may 
be formed there from the components. Into such matters 
the theory of phase equilibria does not’enter. It is con- 


A 





B b C 


FiaureE 42. 


102 Three-Component Systems 


cerned only with the limits of such curves and not at all 
with their shapes. 

Between the temperatures 4, and 7 there is a similar 
appearance of a heterogeneous area extending from A, and 
at 7 the two meet as shown in Figure 43. 

This is the only temperature at which the two solids 
A and C can coexist with a liquid which is the mixed 
melt of these two components only. At lower tem peratures 
they can exist only 
with liquids contain- 
ing Bas well. Hence 
subsequent iso- 
therms are of the 
type of Figure 44, 
taken a little below 
the temperature B,, 
where therefore there 
is the further change 
of a new region Bde 
of complexes yielding 
solid B and solution. 

Thereare now three 
areas all compositions in which give heterogeneous or two- 
phase systems, namely, Cab, dac and Bde. Complexes 
falling in the area Aac, which is a portion of one of the uni- 
variant spaces previously described, resolve themselves into 
three phases, namely, solids 4 and C and the liquid a. 
‘Three phases possess two degrees of freedom. In Figure 44 
both have been exercised in settling pressure and tempera- 
ture. The system is therefore (isothermal, condensed) 
invariant. So long as temperature and pressure are main- 
tained constant, the compositions 4, Cand a are invariable. 
Complexes falling in the area alede remain completely liquid. 








Figure 43. 


~ 


Three-Component Systems 103 


At still lower temperatures, ¢ and d approach and meet 
at p (Figure 41): ¢ and 0 at g (ibid.). Below these respec- 
tive temperatures the changes are similar to those which 


occurred below 7 in 
Figures 41 and 48. 
Hence, an _ isothermal 
section below both p and 
gq, but above #, has the 
form of Figure 45, 

No binary mixture 
ean any longer be liquid. 
The liquid ternary mix- 
tures are those of com- 
positions falling in the 
area abc. ‘The two-phase 
(liquid and solid) areas 
are dac, Bbe and Cba 
respectively. The three- 
phase (two solids and one 
liquid) areas are Aac, Bbc 
and Cab, 

The sections at tem- 
peratures lower still show 
a continuation of the 
decreases in the single 
and double phase areas. 
At # these have shrunk 
to a point and lines re- 























B 


Figure 45. 


spectively, and below / they have vanished: solids alone 


may exist. 


Changes with alteration of composition. The changes 
which occur as the composition of a complex is steadily 
varied may be read immediately from any of these iso- 


104 Three-Component Systems 


therms. An example will be taken from Figure 45. X is 
a mixture of solid 4 and of solid C, containing CX per cent. 
of A and AX per cent. of C. Successive additions of B to 
it give complexes represented by points along the line XB. 
On the first addition of B, the stable phases are liquid a 
and the solids 4 and C, portions of which must therefore 
— melt to form with B this liquid. The proportion of liquid 
steadily increases as the amount of B increases. When 
the mean composition of the system is that of the point P, 
the fraction PX’/aX’ will be liquid of composition a, and 
the remainder, aP/a.X’, will be what remains of the original 
solid mixture of 4 and C, now containing only CX’ per 
cent. of A, but AX’ per cent. of C. Thus, in this particular 
case, more A than C’ has dissolved on addition of Bb. The 
relative amount of 4 in the solid continues to diminish 
until at 2 it becomes zero. The system then gains a devree 
of freedom, changing from (isothermal, condensed) in- 
variance to univariance. The composition of the liquid 
may be varied while the equilibrium remains one of two 
phases, liquid and C. Thus, between z and y the system 
contains a liquid of concentration steadily changing from 
ato y, and a solid which is pure C. The amount of solid 
decreases as B is added, the proportions between solid and 
solution at any stage being immediately ascertainable. 
Thus, at d, ed/eC of the total is there as solid C; dC/C as 
liquid of composition ¢. At y the solid phase vanishes, 
and any more # added dissolves completely, there being 
but a single liquid phase up to the point 7 This point 2, 
an (isothermal, condensed) univariant system, represents 
the limit of solubility of B, and any more added merely 
remains as solid. The changes have therefore been from 
a two-phase (solid-solid) system at X to a three-phase 
(solid-solid-liquid) along Az; thence to a two-phase (solid- 


Three-Component Systems 105 


liquid) along zy and to a single-phase liquid along yz, 
ending at last in a different solid-liquid system along 
wB, 

Compound formation between components, It is seldom 
that a ternary system exhibits the simplicity of Figure 41. 
Cases of it are found with metals, a good example being 
that in which the components are lead, bismuth and tin. 
But, as a rule, one finds the added complications that come 
from the formation either of compounds or of mixed 
crystals between some or all of the components. So many 
_ highly important systems are of the kind where compound 
formation occurs, that typical cases of some consequential 
modifications of Figure 41 and of selected isotherms will 
now be considered in detail. 

As the simplest example, one may take a case where two 
of the components, say 4 and B, are able to form a single 
compound, 4,,8,. Figure 46 shows one of the resulting 
modifications of the three-dimensional model. In this it 
has been assumed that the compound 4,,, is stable 
throughout the whole temperature range. 

A,, B, and C, are, as before, [the respective melting- 
points of the pure components; H, that of the compound 
A,,6,. The form of the curve on the limiting plane 
above’ AB is that of a two-component system already 
exemplified in Figure 26 (Chapter IV) showing two 
eutectics. 

At 7, the solid phases are 4,,B, and 4; at r,, 4,,B, 
and £. Addition of C lowers the temperatures of both 
equilibria ; the former to /,, where it meets the curve gZ, 
of the lowering of eutectic 4—C by addition of B; the 
latter to 1, where it meets pH,. H, and EF, are therefore 
both ternary eutectics, the systems of solid phases being 
A—C—4d,,b, and b—C—A,,B,, respectively. 


n 


106 Three-Component Systems 


These systems of three solids and a liquid do not possess 
any degrees of freedom when pressure has been fixed. 
When, therefore, temperature is altered, one phase must 
go. The equilibrium will be that of an equation showing 











(J. Temperature —> 
\ a oo 





FI@URE 46, 


three solids on one 
side and the liquid 
on the other. 
Liquid is produced 
when heat is added 
so that one solid 
disappears. Hence, 
three curves (each 
giving = composi- 
tions of liquids co- 
existing with two 
solids) must pro- 
ceed upwards from 
each of the points 
#, and &#, One 
of these in each 
ease will be for 
the system 

C—A,,B,,—liquid, 
and the two must 
form a continuous 
curve. In other 
words. the curve 


connecting /, and #, in Figure 46 must pass through 
a maximum as shown. It cannot run directly from one 


eutectic to the other. 


Isothermal sections of Figure 46. Kven though this solid 
figure is very like Figure 41, it is worth while to show the 
precise types of sections of it taken at different levels of 


Three-Component Systems 107 


temperature. At that marked I the type* is that of 
Figure 42, but at all lower levels the compound 4,,8B,, 
gives rise to differences. Figure 47 is the isotherm at the 
level II. All mixtures 
are liquid except those ~ 
which fall in the areas 
Aaband cde. Theformer 
separate into solid A 
and a solution some- 
where on ad: the 
latter into solid 4,,8B,, 
and a solution on ced. 
Lowering of tempera- 
ture increases the areas 
of heterogeneity and 
a section at 7, (Figure 
46) would show a and 
e coinciding. Below 
this, as at the level 
IIT, the isotherm is of 
the type of Figure 48. 
A solution of com- 
position e, which is on 
the curve 7,f, of 
Figure 46, is in equi- 
librium with both solid | 3B 
Popandsd 2B. Any 
mixture of 4, 6 and 
C, having a composition falling in the triangle dé, will at 
equilibrium consist of solution e and a mixture of solids 




















Figure 48. 


* In these sections no attempt is made to retain the relative pro- 
portions of the various regions of Figure 46: illustration of types only 
is aimed at. 


108. Three-Component Systems 
A and A,B No mixture of these solids will melt to 


m-~ 1° 

give a liquid phase alone. de is the curve of solutions that 
can exist with 4,,B,, 
A eb of those with A, and 
| gh of those with solid .B. 
At7,, g and d coincide. 
At C,, and below it, 
solid C appears. At 
the level marked IV, 
the section is as shown 
in Figure 49. Any 
complex falling in the 
new triangle B4f gives 
solution of composition 
FieurE 49. & and a mixture of 
solids 6 and 4,,86,. 
The area of hetero- 
geneous phases 4 and. 
solution, deb, is be- 

coming smaller. 

At g, 2 and J co- 
incide. Below g, as 
at the level V, the 
isotherm type is given 
by Figure 50. 

Liquid binary mix- 
tures of A and C are 
no longer stable. Any 
complex lying within 
the triangle 4vzC consists of liquid of composition 2 and 
a mixture of solids d and C. The region of existence of 
unsaturated liquid mixtures has been reduced to that 
enclosed by Akenm. 'This shrinks still further as the 




















Figure 50. 


Three-Component Systems 


109 


temperature falls. 4% and m steadily approach one another 
At the temperature of p, 2 and m 


and so do e and x. 
coincide, and_ liquid 
binary mixtures of B 
and C cannot exist 
below it. 

Below p, but above 
f,, the section (VI) 
euts across four satu- 
ration surfaces and 
there are four (iso- 
thermal, condensed) 
invariant points, 4, e, 
m and 0, as seen in 


Figure 51. 
The region keno of 
unsaturated liquid 


phase is shrinking, but 
with further fall of 
temperature it does not 
decrease steadily to 
zero. Instead, the two 
curves fe and 20 ap- 
proach one another and 
at the level of VII 
(Figure 46), which is 
the maximum point of 
the curve £,/,, they 
touch as shown in 
Figure 52, giving a 


single solution, S, saturated with both 


(AmBn)f 





























Figure 52. 


C and 4A_B. and 


mn 


dividing the area of unsaturated liquid into two parts, ens 


and fos. 


110 Three-Component Systems 


Then below this maximum point, but still above both 
EF, and #,, at, say, the level VIII, this separation becomes 





> 


eee 


Fiaure 53. 


A 





‘ 
. 2 
. 
x 
eS 
s S 








Fiaurn 54. 


complete, the  iso- 
therm taking the 
form of Figure 53, 
which may be divided 
into two similar por- 
tions by the line C/. 

Comparing this 
with Figure 51, 
which is section VI 
of Figure 46, it will 
be seen that the curve 
ke in the latter is now 
represented by only 
the two end portions 
Au and e¢: similarly 
on by ow and at. 
What were three 
areas have thus now 
been divided into. 
six by the inter- 
position of a seventh 
area fuCt, complexes 
falling within which 
resolve themselves 
into solid 4,,6, and 
solid C with one or 
other of the solutions 


‘wor f. 


At H, (Figure 46) e, ¢ and ~ of Figure 53 meet in a 
point, the two-phase areas den, fe¢ and Cnt diminishing 
to lines. Thereafter the solid phase A cannot exist in 


Three-Component Systems 111 


equilibrium with any liquid, and the isotherm is that 
shown in Figure 54 (level IX in Figure 46). 

At #, the limit of existence of any liquid phase is similarly 
reached, the isotherm containing just a single point. All 
ternary complexes at this temperature will give a solution 
of the composition of this point together with the three 
solids B, Cand 4,,B,. 

The precise description of the boundaries of all the poly- 
phase systems in the prism of Figure 46 should now not 
present any difficulty. 

Modifications of Figure 46. Besides this case, where the 
compound 4,,B, is stable throughout the whole tem- 
perature range, there are two other possibilities to which 
it is worth while to draw attention, without staying to go 
into detail. The one is exemplified in Figure 55 and 
shows a lower temperature limit, 7 to the existence of the 
compound. The other, Figure 56, shows a case where the 
compound only comes into existence in. the lower part of 
the temperature range, at a point also marked 7’. 

In both cases we have at 7’ a (condensed) invariant 
system of four phases, three solids, 4, 6 and 4,,B,, and 
a liquid, and it is easy to see that the reactions at these 
points may be represented by the equation 

A+ B= A,B, + liquid. 

In Figure 55 rise of temperature causes the action to pro- 
ceed from left to right with disappearance of A or B: in 
Figure 56 from right to left, with disappearance of 4,,B,,. 
The liquid phase does not disappear in either case what- 
ever the direction of the temperature change, so that 
7 is a transition point between solids and not a eutectic 
point. 

Common projections. While it is very advisable for a 
student to study with the aid of space models or perspective 





Three-Component Systems, 





Fieure 56. 


drawings the principles 
underlying these ternary 
systems, there are occa- 
sions when it may be 
more convenient to use 
projections. The two 
in most common use are 
the orthogonal projection 
downwards on the tri- 
angular base, and the 
perspective horizontal 
projection on to one side 
of the prism from the 
opposite vertical edge. 
The first method is used 
very often by Schreine- 
makers, and an example 
is seen on the base 
triangle of Figure 41. 
The significance of the 
three curves and the 
four eutectic points 
(three binary and one 
ternary) is obvious. No 
information regarding 
temperature is given, 
though the direction 
in which temperature 
increases as concentra- 
tions alter may be, 
and usually is, shown 
by arrows, - 

The second method is 


Three-Component Systems 1138 


illustrated in Figure 57, which gives the projection of 
Figure 46 on the vertical plane B, BAA, from successive 
points along the line CC, and upwards. 

It is advisable for the student to work out for himself 
the limited significance of the divisions in the projection 
as compared with those of the full Figure 46. Janecke 
first suggested this type of projection. It can give only 
a portion of the information contained in the space model, 
but is very useful for tracing the courses of certain changes 
in phases with alteration in 
temperature. From the present 
point of view it is not necessary 
to consider it further. Bi 


At 


The discussion of these re- 
latively simple cases may be 
taken to indicate the lines upon 
which still more complicated 
systems than those of Figure 46, 
&e., may be worked out. The 
general treatment is similar when 
A and & ean form more compounds 
than one and when # and C, or C and 4, or both, form one 
or more compounds, or when ternary compounds may exist. 

The literature of the subject abounds in illustrations 
incompletely worked out. It is seldom that more than 
a few isotherms are determined: two or three usually 
suffice to give a reasonable lead as to the form of a required 
part of the complete three-dimensional figure. In parti- 
cular, a great deal of attention has been devoted to cases 
where one component is a salt and the other two are 
solvents, or two are salts and the third a solvent; and such 
investigations are usually limited to temperature ranges 








Eemperatung ak 


2 





B Composition A 


FIaureE 57. 


2568 H 


114 Three-Component Systems 


above the freezing-points of the solvents. It is perhaps 
worth while to consider a few isolated isotherms of this 
kind and to take the opportunity to discuss certain practical 
points in the use of triangular diagrams. 

Some typical isotherms: solvate formation. Figure 58 is 
an example of a system in which the salt S is able to 
form with one solvent W (say water) several compounds 
(hydrates) of compositions represented at f,, H, and /7/,, 
but is unable to form 
a compound with the 
other solvent A (say 
aleohol). * 

There are four distinct 
solubility curves. Solu- 
tions along the first, ad, 
are saturated with respect 
to hydrate H,;_ those 
along 4c with respect to — 
ff, ; cd to H, and de to 8. 
6; c and d are solutions in 
equilibrium with the pairs of solids //, and H,, H, and A,, 
and H., and S, respectively. From the diagram one ean tell at 
once how any mixture of the three components will resolve 
itself at equilibrium, and in illustration one may take the 
changes which will occur when solvent A (aleohol) is added 
in successive amounts to the hydrate H,. The average 
compositions of the mixtures so formed will lie along the 
line //,A. 

On first addition of A, dissociation of some HH, occurs, 
giving rise to solution of composition 4 and formation of 
some of the lower (that is, less hydrated) compound /,. 
This three-phase system will persist, with steady increase 
in amount of /H, and decrease of //,, until the point / is 





Fiaure 58, 


Three-Component Systems 115 


reached where the line 17,4 cuts 7,4. Here the amount 
.of H, has dwindled to zero and the system is one of two 
phases, solution 4 and #,. This solid phase persists as 
more alcohol is added, but the solution steadily passes from 
composition 4 to ¢. Beyond the intersection g, the third 
hydrate, #,, makes an appearance, and between g and / the 
three phases //,, 7, and solution ¢ are in equilibrium, the 
amount of H, steadily increasing as //, diminishes. At 
h, H, has vanished, and 

between 4 and y, ff, 

exists with solutions 
continuously changing 

from c tod. Between s 
y and f#, the third 
three-phase = system 

occurs, the solution . 
being d and the solid i 
H, and the anhydrous 
salt. At the former Solv-t Solv. 2 
vanishes, and from & to wee 

m the latter is in equilibrium with successive solutions 
from d to m. At m the salt has entirely dissolved and 
further addition of alcohol gives unsaturated solutions only. 

In following this sequence of changes one has dealt of 
course only with systems that are in stable equilibrium. 
If the attempt were made to realize this sequence experi- 
mentally, metastable conditions would certainly be passed 
through before the final states were established. 

When both solvents are able to combine with the salt 
chosen as third component, the form of the isotherm is that 
shown in Figure 59, if only one solvate exists in each 
ease, This offers nothing new (from the point of view of 
interpretation) as compared with Figure 58. 

H 2 


S 











116 


Three-Component Systems 


In both these casesthetwo solvents have beensupposed mis- 


cible with one another in all proportions,and at the tempera- . 


ture selected there hasnot beenany separation of a liquid phase 

















Ba 





into two immiscible liquids 
of different compositions. 
Instances occur where 
neither condition is fulfilled. 
* Partially miscible liquid 
components. Figures 60 
and 61 illustrate a case 
where component A is 
soluble in each of the 
solvents £ and C, which 
are only partially miscible 
with one another. 

In Figure 60 a and 


the two layers formed 
by mixing # and C in 
any proportions given 
by points along ad. 
Addition of 4A may or 
may not cause the 
layers to approach one 
another in composition. 
It has been assumed in 
b C the figures that the 
miscibility does in- 


crease as A is added, giving steadily approaching: layers, as 
at c and @, which become identical at ¢. ./g, lying above 
the curve acedd, gives solutions in equilibrium with solid 4. 

Figure 61 represents the same system at a different 
temperature where the curves intersect. There is no longer 


6 are concentrations of — 


* 


Three-Component Systems 117 


a continuous series of solutions in equilibrium with solid 
A, but at ¢ a second liquid layer of composition @ appears, 
the two being able to exist in equilibrium. All mixtures 
of average compositions represented by points within the 
triangle decd will resolve themselves into solid 4 and 
these two liquid layers. Thus, of a complex o, the frac- 
tion op/pd will be solid A, and od/pd will be in the 
liquid layers, of which the proportion dy/de will be in 
the layer of com- 
position c, and 
epfed in that of 
composition d., 
Complexes beyond 
Ac and Ad will give 
solid 4 and single 
solutions. Those 
within acd) will of 
course give two 
liquid phases and 
no solids, and a 
fuller treatment of 
such cases will be Figure 62, 

given presently. 

Partially miscible ternary liquids. A somewhat similar 
case to this may occur even when the two solvents are 
miscible in all proportions. At certain temperatures and 
in the presence of certain proportions of the third com- 
ponent A, two liquid phases may appear. Figure 62 shows 
in perspective the concentration-temperature space model 
of such a system. 

In the lower portions, as at J, the isothermal section is 
similar to the upper part of Figure 60. At / the first 
sign of a separation of the solution into two layers appears. 





118 Three-Component Systems . 


At higher temperatures, as at I//, the section is that of 
pene 63. The regions Afe, ded and Adg do not differ 





A 
yi 
fe . 
e 
B 
FIGureE 63. 
A 





B 


Figure 64. 


composition of this 


from those similarly 
lettered in Figure 61 ; 
but with increase in 
the amounts of 5B and 
C the two liquid layers 
approach in concentra- 
tion and ‘ultimately 
coincide at #. 

Other common types 
of isotherm, -A system 
of one solvent B and 
two soluble = com- 
ponents 4 and C, each 
of which may form 
a solvate (S4 and Sp), 
and which may unite 
to form a molecular 
compound (S4¢), such 
as a double salt, is 
illustrated in the iso- 
therm, Figure 64. 

Where a_ ternary 
compound may be 
formed the diagram 
includes a portion such 
as abe in Figure 65, 
where c gives the 

Complexes in the 


triangle AcC resolve themselves into solids only, namely, 
A, C and the ternary compound, in proportions at once 


calculable, 


Lhree-Component Systems 119 


Finally, in Figure 66 is an actual case in which the 
components-are water, sodium oxide and chromic anhydride 
at a temperature (20°C.) where no less than eight different 
solid phases may exist according to the proportions between 








Figure 65, 


Na,0 


(a) NaOH. H,0. 

(6) Na,O. CrO,. 

(c) 3Na,0. CrO;,13H,0. 
(d) Na,O. 4Cr0O,.4H,0. 
(e) Na,O. 2CrO,.2H.0. 
(f) Na,O. 3CrO;. H,0. 
(g) Na,O.4CrO,.4H,0. 
(h) CrOz. 


10s 








Figure 66. 


these components (Schreinemakers, Zeit. phys. Chem., 1906, 
55,71). This diagram will afford good exercise in inter- 
pretation in the light of the general discussion which has 
been given. , 

Determination of compositions of solid phases. o arrive 
at a knowledge of the exact solid phase, or phases, in 


120 Three-Component Systems 


equilibrium with a particular solution is an essential part 
in the determination of such systems as these. After 
separating the solid by filtration, or other means, it is 
often a troublesome matter to free it completely from 
adhering solution, keeping the whole exactly at the tem- 
perature of the isotherm sought. 

A simple graphical procedure, dependent upon the use 
of conjugation lines, enables one to avoid the necessity for 
this. The method may be illustrated from Figure 67. 

a is a solution in 
equilibrium with a 
solid of unknown com- | 
position. By filtration, 
a portion of moist 
solid, that is, solid and 
adhering solution, is 
removed. Devices for 
maintaining constant 
temperature during 
filtration will be neces- ~ 
sary or not according to 
the temperature of the 
isotherm and the rate at which filtering can be carried out. 
A known weight of moist solid, or residue (German Rest), 





FIGURE 67. 


is then analysed. Since it is a mixture of the solution — 


a and the pure solid, its composition, represented at a’ in the 
figure, must lie somewhere on the straight line connecting 
those of solution a and of pure solid. The more completely 
adhering solution has been removed in the filtration, the 
nearer will it lie to the latter, The composition of the 
solid is therefore somewhere on aa’ produced. By an 
exactly similar argument, this composition will be on 
6)’ produced. The intersection of the two lines, the point 


Three-Component Systems 121 


S, therefore gives at once the desired value, and other 
solutions and residues will confirm, or correct, the point. 
Thus one arrives at a knowledge of the solid phase without 
requiring to obtain it perfectly pure for analysis. The 
procedure may be termed the Residue (or /test) Method. 

If the chemicals dealt with are quite pure, there is no 
occasion even for an analysis of a residue. A complex, or 
- mixture of components, in exactly known proportions 1s 
prepared and allowed to pass to the equilibrium condition. 
Then, referring again to Figure 67, if c’ be the mean 
composition of the complex and ¢ that of the solution 
formed, the composition of the solid phase must lie on 
ce’ produced. From a similar set of figures for another 
complex and solution, an intersection giving the composi- 
_ tion of the solid is obtained just as before. 

- This method is rather simpler than the preceding one, 
but in practice it is seldom preferred because usually it is 
advisable to take the complex fairly near to the solution 
point, so that the extra-polation becomes a lengthy one, 
and errors in composition of either solution or complex are 
magnified in that of the solid. Nevertheless it might be 
employed with success much more than is done. 

Intersection of solution curves. It is to be pointed out, 
in passing, that all complexes, or residues, lying in an area 
like the triangle SdC of Figure 67, will resolve themselves 
into a solution and a mixture of solids. Thus d’ gives the 
fraction ed’/ed of the total as solution d, while dd’de 
consists of solids of which Ce/CS is the double salt S and 
Se/SC is the component C. When two complexes, as 
at d’ and d”, give the same solution, as at d, it is evident 
that d is a point of intersection of two solution curves. 

Practicat points in preparing pure solid phases. When, 
however, it is desired for some reason to obtain a pure 


122 Three-Component Systems 


solid, say a pure double salt such as J, or J, in Figure 68, 
free from admixture with the solution with which it is in 
equilibrium, one can often obtain from an isotherm useful 
guidance as to procedure. 

Suppose B is a solvent, such as water, 4 and C being 
salts capable of forming the double compounds D, and J. 

Filtration of a complex containing solution and either 
of the double salts will give the solid with more or less 
adhering solution. The natural next step might be to 


wash the crystals. 


rapidly with water, 
then removing residual 
water as quickly as 
possible by suitable 
means. Provided that 
these operations can be 
earried out so fast that 
no progress towards a 
new equilibrium «is 











be satisfactory. Other- 
wise the result will 
vary according to the relative positions of the curve giving 
solutions in equilibrium with the double salt, the point 
corresponding with the composition of the double salt, and 
the angle representing pure water. Thus in Figure 68 if 
impure D,, of mean composition given by a point just inside 
the triangle D,ae, be treated with water so as to give 
a complex of mean composition falling as at ¢ within the 
triangle DjeD,, a change will oceur, and will become more 
or less complete according to the time available, giving 
solution e¢ and a mixture of the solids, D, and D,. In 
other words, the double salt D, is decomposed by water and 


FigurE 68. 


made, the result will 


(a, 


Three-Component Systems 123 


the latter should therefore be avoided for purposes of wash- 
ing. On the other hand, the line joining D, or a point 
near it cuts the curve eb, and therefore a complex, 
such as d, cannot give rise to any other solid. Washing 
with water will therefore be less objectionable in this 
case. 

Finally, there may be cases where it is advisable to allow 
_ complete equilibrium to be attained in the complex formed 
by addition of water to one substance 4 which is con- 
taminated by another, C. Such contamination may have 
been brought about by 
overcooling a solution de- 
positing A to a point at 
which both 4 and C may 
deposit together. In Figure 
69, which is an isotherm for 
water (4), A and C at about 
ordinary temperature, the 
point I represents (to an 
exaggerated extent for the 
purposes of the diagram) Figure 69, 
such a product. 

Addition of water to I/ gives a complex represented at a, 
say, and this resolves itself into solution 4 and solid J’. 
The latter is very much nearer to pure 4 than J/, and 
repetition of the same procedure will give a product J/” 
still closer. Moreover, a knowledge of the proportions in 
which J/ and water are mixed will enable one to calculate 
at once from the diagram the ‘ yield’ of J/’ and the quantity 
of solution 6 formed. Such a process of ‘steeping’ or 
‘soaking’, if practicable, is very much to be preferred to 
what is ordinarily termed ‘ purification by recrystallization ’ 
and which involves first the complete solution of the whole 

















124 Three-Component Systems 


impure mixture at a suitable temperature, and then separa- 


tion of the pure solid by cooling.” Especially in large-scale ~ 


work is the avoidance of these operations desirable. 














Figure 71. 





It may, however, 
happen that ‘steep- 
ing’ will lead to 
little or no puri- 
fication, or even to 
a relative increase 
in impurity; and 
again the required 
information may be 
obtained from a 
suitable isotherm. 
Thus in Figure 70, 
if M be steeped 
with water in pro- 
portions giving: the 
complex a, there 
will be formed solu- 
tion 6 and a solid 
M’ hardly differing 
at all from 1/7: while 
in Figure 71, steep- 
ing the impure pro- 
duct JJ may lead 


to M’, containing 


actually an increased percentage of the impurity C, In such 
a case purification by steeping is no longer practicable. 
*Congruently’ and ‘incongruently’ saturated solutions. It 


is pertinent at this point to return to Figure 68 in order 


to draw attention to the presence of two very different types 
of (isothermal, condensed) invariant systems, 


Three-Component Systems 125 


There are three invariant points in the figure, namely, 
a,eandd. Ifa conjugation line be drawn from & through 
each of these points to the side AC, it is at once clear that a 
and 6 are similar systems, while e belongs to a different 
type. The differences appear in two respects. In the 
first place, the composition of solution a (or, mutatis 
mutandis, of 6) may be expressed in terms of the solvent B 
and a mixture a’ of the solids, 4 and /,, with which the 
solution is in equilibrium. In the next place, evaporation 
of B from a will give a complex of mean composition lying 
on the line aa’, which means that 4 and JD, will both 
separate out during the process. Such a solution is com- 
monly said to be ‘congruently’ saturated with respect. to 
the two solids. 

By contrast, the composition of e cannot. be expressed in 
terms of the two solids, D, and D,, with which it is in 
equilibrium, and the solvent B. No line connecting 6 and 
any point on D,), passes through e. Again, evaporation 
of B from e gives a complex lying in the area aeD,, not in 
D,eD,. The solid separating will then be D, alone, not 
a mixture of both the solids with which e was saturated. 
The reaction occurring during evaporation results in the 
disappearance of D, as a solid phase (if any be present 
initially with ¢), with production of solution lying along: ea. 
Such a solution as é is said to be ‘incongruently ’ saturated 
with respect to the two solids. These terms are in very 
general use. 

General case of partial miscitility of one parr of liquid 
components. In the examples so far treated, the general 
type of equilibrium has been that between one or more 
solids and a solution. It was convenient to illustrate in 
Figures 60, 61, 62 and 63 the modifications introduced into 
particular isotherms when two liquid phases appear, but 


126 Three-Component Systems 


the general case of partially miscible liquids requires rather 
fuller treatment. It will suffice to consider a selection of 
the possible cases, 
taking first those 
where one, then those 
where two, and, lastly, 
those where all three 
pairs of liquid com- 
ponents are at certain 
temperatures only 
partially miscible. 
Figure 72 is an ex- 
ample of the first 
kind. 

A and B are par- 
tially miscible; 6 
and C,and A and C, 
wholly. All — iso- 
thermal sections be- 
low & are of the type 
of Figure 73. 

Here a and @ are 
the two liquid layers 
formed by any binary 
mixture of 4 and 6 
in proportions repre- 


s sented by points lying 


Temperature —» 





on the line ad, Mix- 
Figuae 72. tures falling on Ba 

or 6A are homoge- 

neous or single-phase. Such cases of binary systems in 
two liquid layers were not discussed in Chapter IV, to 
which they properly belong: the present brief account of 








Three-Component Systems 127 


ternary systems, however, will incidentally give a sufficient 
account of the simpler ones. 

Critical solutions: binodal curves and surfaces, Addition 
of C brings about an approach in composition in the two 
layers, as shown at d and e, or f and g. Finally, at a 
point c the two layers become identical; 4 has travelled 
along the curve Jege and a along adfc. The whole curve 
begcefda is commonly termed a ‘binodal curve’. Any 
complex of mean composition represented by a point 
within the area cut off by it and ad will separate into 
two layers the compositions of which one may represent, 
as in the diagram, as connected by conjugation lines, or as 
one sometimes calls them, tie-lines. ¢ is a critical point at 
which the two layers merge into one, and the solution 
at it is called the ‘critical solution’ for the temperature 
considered. 

At higher temperatures, if mutual solubility be assumed 
to increase as temperature rises, the isotherms making up 
Figure 72 are of similar form, but the length of the binodal 
curve steadily decreases and ultimately it becomes the 
point 4. The eurve cé connects all the critical points of 
the intermediate temperatures and is usually termed the 
‘critical curve’: the binodal curves make up the ‘ binodal 
surface’ acbk. & is the upper critical point of the system : 
at all temperatures above it every binary or ternary mixture 
of the components forms a single liquid phase. The system 
in Figure 72 shows / on the binary surface A467’; but of 
course it may happen that the upper critical point 
is attained in a ternary, instead of in a_ binary, 
mixture. Moreover, since there are binary liquid mixtures 
which show increasing miscibility as temperature falls, 
a lower critical temperature in either binary or ternary 
systems, or both, is also possible. Figure 74 shows 


128 Three-Component Systems 


a case where there are upper and lower ternary critical 
points. 
abed is the binary 
system on the- plane 
ABT, a and e¢ being 
the upper and lower 
binary critical points, 
respectively. 4% and #’ 
are the upper and lower 
ternary critical points, 
and akc is the eritical 
eurve. An _ isotherm 
between a@ and 4, or - 
between ¢ and #’, will 
have the form shown 
in Figure 75. With- 
B Cc in these temperature 
ES AT ranges the three com- 
ponents are completely 
miscible in pairs, but 


Temperature —+> 


Temperature —> 





A 


ternary systems of 
compositions falling’ 
within a certain area 
efgh resolve themselves 
g into pairs of liquid 
- phases, such as f and 


g, shown connected by 


/ tie-lines. 
wh \ Relations of this 


B d C kind might even hold 
Fieure 75. . 

for all temperatures in 

the range of partial miscibility: im such eases there 

would be, at any one of these temperatures, two critical 


Three-Component Systems 129 


solutions, as ¢ and /, and the complete critical curve would 
be a closed one. 

Applications in quantitative analysis, Two applications of 
the relations represented in Figures 78 and 74 to analysis 
of mixtures are well worth noting. Suppose that B and C. 
in Figure 76, are miscible in all proportions, and 4 and C 
likewise, but that 4 and B are only partially so. The 
binodal curve for a convenient temperature of working 
is acb, The practical methods of obtaining these curves in 
the first place must be 
sought in more detailed 
text-books. 

If now an analysis be 
required of a mixture of B 
and C (where, for example, 
& may be water and C 
alcohol) in unknown pro- 
portions, the following 
procedure is possible. 

Liquid 4 (toluene, for 
example) is added gradually 
to a known weight W of the mixture. At first it dissolves 
completely on shaking. Then a point is reached at which 
a faint turbidity remains permanently, the amount of 4 
which has been added being W’. This turbidity is due to 
the appearance of small particles of a second liquid phase. 
Addition of more A will cause the separation of more of 
this phase, giving, ultimately, two distinct layers. When 
the first turbidity appears, the percentage of A in the 

/ 








Fiaure 76. 


: i ne 
ternary mixture is Wa * 100, represented, say, by the 


length 54 (or C?) in the figure. The exact composition, 
therefore, of the mixture must lie somewhere on 4/, parallel 


2562 I 


130 Three-Component Systems 


to BC. It must also lie on the binedal curve, since a second 
liquid phase has occurred. It must then be at the inter- 
section 7. It follows at once that the composition of the 
original binary mixture is given by g, the point of inter- 
section of the line Af produced with the side BC. The 
accuracy of the determination, which is really a titration, 
may be tested by 
continuing the addi- 
tion of A until a 
point is reached at 
which the two liquid 
phases again be- 
come one. If the 
percentage of 4 in 
this final mixture 
be given by Jd, 
and if dA, parallel to 
BC, cut the binodal 
eurve at ¢, then 4, 
e and f# must le on 
the same straight 
line. 

For the analysis 

Riche yd) of mixtures lying 

nearer to C than 

the intersection with BC of the tangent from A to the 

binodal curve, it is obvious that a known amount of pure 

B must first be added before the titration method can be 
applied. 

A second principle which has been applied for pur- 
poses of analysis may be briefly illustrated from such 
a case as that of Figure 72, which is reproduced, some- 
what modified and with additions, in Figure 77. & gives 


Temperature —> 





Three-Component Systems 131 


both the temperature and composition of the binary 
critical solution of 4 and B. 4%’ is the ternary critical 
solution curve when a third component C is introduced. 
kk’ is the vertical projection of 4% on the base tri- 
angle. Only those ternary mixtures having compositions 
lying along #’%’ can have critical solution temperatures. 
For all others in the heterogeneous area there will be a 
temperature at which homogeneity occurs, but this will be 
due to the decrease in amount, and ultimate disappearance, 
of one of the two layers, not to the merging of finite 
quantities of the two because of identity of composition ; 
a totally different phenomenon. In the one case the pas- 
sage is to one end of a tie-line: in the other, the two ends 
of a tie-line coincide. 

Suppose that an impure sample of 4 contains an unknown 
proportion of C, which it is desired to determine. Its com- 
position will lie somewhere on AC. Now the temperature ¢ 
of the critical phenomenon between this mixture and B 
may be experimentally determined. The procedure will 
involve progressive additions of 6 to the mixture until the 
proportion is reached which at the right temperature gives 
the characteristic critical change from two layers. The 
composition of the mixture will obviously le at that point 
k, where the ternary critical curve cuts the isothermal 
(¢ deg.) triangular plane, or, what comes to the same thing, 
at a point 4,’ in the base (¢ = 0) plane which is at a dis- 
tance ¢ vertically below 44’. If now a horizontal line be 
drawn from & (or the corresponding point at temperature ¢) 
through 4,’ (or &,) and produced, it will cut AC (or the plane 
ACt) at a point wx, (or #,) which gives the previously un- 
known proportion of Cin A, Similarly any other mixture 
@,', #3; may be analysed. 

The method is of course quite empirical, since one first 

| 12 


132 Three-Component Systems 


determines 44’ for known mixtures of 4, B and C, and then 
uses this knowledge conversely to ascertain unknown con- 
centrations. Its sensitiveness depends upon the sharpness 
of fall of the curve £2’, that is to say, on the magnitude of 
kk,as compared with 4v,’.. In many cases the temperature 
- gradient is very steep indeed, and quite small additions of 
C to A make large changes in the critical solution tempera- 
ture. 

As a rule the method is applied for the detection of only 
small proportions of C in A, so that it is necessary for the 
analyst to know only a part of the ternary critical solution 
curve. More often than not, investigators simply plot 
a few alterations in critical solution temperature against 
the corresponding percentages of C in the 4—C mixtures, 
and use this curve for the interpolations. It is necessary, 
however, to be cautious when only a partial investigation is 
made, since in such a case as that of Figure 74 there will 
be two quite different mixtures of 5 and A containing C 
which will give the same critical solution temperature. 

The objection to using changes in critical solution tem- 
peratures as criteria of purity of one of two components is 
that they so often do not fall in convenient ranges of 
temperature. This may be avoided by using a ternary 
mixture of composition selected to give a critical solution 
temperature at a convenient point. Such a temperature 
will similarly be affected by additions of a fourth component 
as impurity in one of the three, and a relation between the 
temperature changes and composition may be experimentally 
established. 

Partial miscibility of two paws of liquid components, 
Passing now to the general case where at certain tempera- 
tures partial miscibility is characteristic of two of the three 
pairs of liquid components, a couple of space models will 


7 


Three-Component Systems 133 


serve as illustrations. The simplest case will be that in 
which both pairs of partially miscible liquids have upper 
critical points, and where, at a temperature below both of 


these, the form of the 
isotherm will be that 
of Figure 78, where 
the two heterogeneous 
regions are separated 
from one another, each 
with its own critical 
solution a or 0. 

At lower tempera- 
tures these regions will 
become greater: at 
some point they will 
touch and then merge 
into one another 
giving isotherms like 
Figure 79, in which 
there is no longer any 
critical solution. 

The building of a 


‘space (composition- 


temperature) model for 
this case does not offer 
any difficulty. 

It may sometimes 
happen, however, that 











Fieure 78. 


A 





FIGURE 79. 


with rise of temperature the heterogeneous region of 
Figure 79 does not separate into two distinct parts. This 
will be the case, for example, if the temperature of the binary 
critical solution point for 4 and £B is reached before such 


separation has occurred. 


The isotherm at this point will 


134 Three-Component Systems 


be as in Figure 80; and thereafter, as temperature rises, as 
in Figure 81 until heterogeneity ceases, 
A typical three-dimensional model including these types 








Figure 80. 





~S 
Temperature —> 





Figure 81. Figure 82. 


is given in Figure 82. Actually it represents diagram- 
matically the system water (4), phenol (B) and aniline (C) 
(Schreinemakers, Zeit. phys. Chem., 1899, 29, 577 and 30, 
460). 


emperature —> 





Three-Component Systems 135 


Here the upper critical points are on the respective binary 
systems, but it is a simple exercise to draw the altered 


diagram for the case 
where one or both of 
the upper points are 
ternary. 

The form of the 
model when one pair 
of liquids has an upper 
and the other a lower 
critical point may also 
be noted. Figure 83 
gives a representation 
of the system water 
(A), phenol (£) and 
triethylamine (C) 
which should be readily 
understood from the 
previous discussion 
(Meerbure, Zeit. phys. 
Chem., 1902, 40, 641). 

Partial miscibility of 
three paws of liqud 
components. Lastly, 
one may indicate by 
a few isotherms some 
characteristic features 
of possible systems in 
which each of the three 
pairs of liquids may 
give two liquid phases 


over certain temperature ranges, 








Figure 83. 


A. 








Ficure 84. 


It will be assumed that 


each pair will be completely miscible at some definite upper 


136 Three-Component Systems 


temperature. Above the highest of these, therefore, if there 
be a heterogeneous field, it must be ternary in composition 
at all points, the isotherm type being that of Figure 75 with 














Figure 86. 





two critical points. 
At lower temperatures, 
one, two or three two- 
phase fields may 
appear, and these may 
or may not have a 
binary system as part 
of the boundary. Ex- 
amples are shown in 
Figures 84 and 85. 
Ultimately, as tem- 
perature falls, every 
field must be bounded 
by at least one side of 
the triangle. Then at 
lower — temperatures 
still, two hetero-- 
geneous fields may 
merge into one, contact 
first taking place at 
the respective critical 
points. Figure 85, for 
example, would then 
pass through the forms 
of Figures 86 and 87, 


where, in the latter, the third heterogeneous field at its 
critical point is just in contact with that formed by the 


merging of the other two. 


At this temperature it is clear that the critical solution a 
is in equilibrium with another solution ¢. Any further fall 


Three-Component Systems 137 


of temperature will now bring the end a of the line da 
into contact with one of the tie-lines in the other area, i.e. 
. a will now be a mixture which will resolve itself into two 


phases lying on the bi- 
nodal curve. This means 
that solution d will be 
in equilibrium with each 
of two other liquid 
phases, which are them- 
selves in equilibrium. 
Thus we attain a three- 
layer system, the type 
of which is shown in 
Figure 88. 

Any complex repre- 
sented by a point within 
the triangle def will give 
three liquid phases of 


compositions d, e and f 


respectively. The posi- 
tion of such a point will 
determine the propor- 
tions of the components 
in each layer. Thus, 
from a quantity of mix- 
ture p, the fraction 
Mp/Mf will go into the 
layer of composition /: 














Figure 88. 


Jp/Mf will be in the other two layers, and of it dM/de 
will be of composition e and el//ed of composition d. 

The three areas shown with tie-lines include all com- 
plexes giving two liquid phases: the remaining three areas 
include the unsaturated or homogeneous solutions. 


138 Three-Component Systems 


These indications suffice to show the relationships that 
are possible in such cases. It will be at once apparent 
that the possible three-dimensional figures may be of rather * 
diverse types according to the natures of the components, 
and the student should amplify the descriptions of the 
various types that have been indicated and deduce for him- 
self other possibilities, 

The equilibrium type: AB+ CD= AD ue CB. One special 
type of three-component system dassree some little atten- 
tion in conclusion, since for its satisfactory representation 
one must employ graphical methods, the consideration of 
which now will facilitate the treatment of four-component 
systems in the next chapter. 

This is a system in which double decomposition may 
occur between two sets of substances. Important and 
familiar examples are the hydrolyses of salts and esters. 
A general chemical equation for reactions of this kind is 

AB+CD= AD+CB. 

It is clear that since the four substances are connected 
by this equation, the composition of an equilibrium mixture 
of the four may be expressed in terms of any three ; since 
if, for example, 4.6, CD and AD are specified, CB is at once 
known from the equation 

ABxCD 
AD 
where & is the equilibrium constant of the reaction, or the 
ratio of the backward to the forward velocity. 

Or, again, one might express all compositions in terms of 
the radicals. Thus if a, 6,c and d are the weights of the 
radicals 4 (+), B(—), C (+) and D (—), respectively, in 
any mixture, the relative numbers of equivalents, @,, 6,, ¢¢ 
and d, are readily calculated, and since they are present in 
such proportions that a,+c, must always equal J,+<d,, it 


= k.OB, 


Three-Component Systems 139 


follows that the value of any one of them is always known 
when the values of the three others are specified. 

Graphical representation. One cannot in such a case plot 
compositions of all possible phases in a triangle such as that 
used hitherto, in which the three components, say 4B, CD 
and AD, are represented by the respective angular points. 
It is impossible, for example, to plot the compound CS in 
such a figure. Some different scheme, therefore, must be 
devised. 

Now since in any particular phase the number of 
equivalents of the positive radicals must equal that of 
the negative, it follows, if we take this number as unity, 
that the composition may be expressed as 


Therefore by plotting w and y as co-ordinates, it should be 
possible to represent compositions of all phases. Rectangular 
co-ordinates will give the simplest case, and the way in 
which the diagram is constructed may be followed by 
considering Figure 89. 

# values are here plotted as abscissae on the axis marked 
X, and y values as ordinates on that marked Y. Ifw=0O 
and y = 0, the phase consists solely of the component 4S, 
that is to say, the point of origin represents pure AJB. 
If x =1 and y= 0, the phase is pure AD, represented 
then by a point unit distance from 4B along the X axis. 
Similarly, for « = 0 and y = 1, we get BC at unit distance 
upwards on the Y axis. Finally «= 1 and y =1, gives 
CD as shown. Thus the four substances taking part in 
the double decomposition may be represented by the 
respective corners of a square. 

Any phase which contains two or more of these four 
substances, and the composition of which is expressed 
in terms of any three of them arbitrarily selected as 


140 Three-Component Systems - 


components, may be represented by a point either on the 
sides of, or within, the square, as the case may be. 

As an example, a phase containing a equivalents of 
AB, b of CD, ¢ of AD and d of BC, which may be written 
as Ai4¢Ba+qlh+qP pee is represented by a point p such 


ate-at Re 
b+ 
iat am ase BF 
ua, eee b+c¢ 
ie oe dae 


since substitution 
of these values in 
c the above com- 


a position reduces it 

| to the general form 
A, i eee 

ao oe ae ade Soe d This expression 

Seaeu in terms of the 


x radicals is more 
ye mS rational, in the. 





a 
AS 


+ 


42) 





REE See Ow aC ce 
—_ 








ms : case of solutions, 

a than any attempt 

sea at expression in 
a 

Ap == terms of the mole- 

Figure 89. cules present, since 


we have no means 
of telling exactly how radicals may pair in a solution, nor 
indeed to what extent they form molecules at all or remain 
separated as ions. In very dilute solutions only can some 
approximate idea be formed. 

Thus the composition of the point p, in Figure 89, may 
be expressed not only, as above, in terms of 45, CD, AD 
and BC, but also as (l—w—y)AB+ybC+a2AD or, 
again, as (l—y)dD+(y—«x2)LC+2aCD, since all three 


Three-Component Systems 141 


statements give the same relative proportions between the 
radicals, namely, those of the general formula, 4,_,,B_,C,D,. 
If, as_in this particular case, y be greater than a, 
the composition cannot be expressed in equivalent pro- 
portions of BC+ CD+AD or AB+ AD+CD without some 
of these substances appearing in negative amount, which 
would be meaningless. These facts are immediately obvious 
from Figure 89, since p, besides falling in the square 
AB—AD—CD-— BC, is also within each of the triangles 
BC —AB—AD and AB— BC—CD; but it is outside of 
both BC—AD—CD and AB—AD—CD, Similarly a point 
p’, for which # is greater than vy, may represent BC+CD+ 
AD in certain proportions, or d4B+CD+ AD, or all four 
AB+AD+CD+ BC, and there is usually no more reason 
for choosing one set of constituents than the other. To 
express all compositions in terms of the radicals is therefore 
the more reasonable method. | 

Representing the relative numbers of equivalents of the 
respective radicals by @ and ¢ (positive) and 4 and d 
(negative) it will be seen that the method really amounts 
to plotting the ratio 

c d 
ao (= y) against ae] (= a). 

If, however, it be desired for any special reason to 
express compositions in terms of molecules rather than 
radicals the proportions may be read just as easily from the 
figure. Thus while p represents 4,455, - a CppDa, it also 
represents ap of AD, bp of BC and pe (= ph) of AB, for, 
since ap = ce, it follows that ap+ép+ype = 1, the side of 
the square. Also it represents ap of CD, pe of AB and pf 
(= pg) of BC, for, since ap = Uf, ap+pe+pf=1. 

A typical ewample. Figure 90 will serve as an illustra- 
tion of a system plotted in this way. The equilibrium is 





142 Three-Component Systems 


AB+CD=AD+CB. The possible solid phases repre- 
sented are CD and a molecular compound K of AB and AD. 
Such a system approximates to that given by the equilibria 
at 30° C, between ordinary alcohol, caustic potash, water 
and potassium ethylate, and one may perhaps most readily 
appreciate the significance of the respective fields from 
a consideration of the changes which occur when, say, 
water (4B) is steadily added to pure potassium ethylate 
(CD). 
BC(cH.0 sHg0K)CD The complexes formed 
” c 7| will lie on the line 
joining CD and AB. 
Pure ethylate may be in 
equilibrium with any of 
the solutions represented 
on the line ef Addition 
of water to it gives a 
‘ complex. falling in the 
oe | triangle of e.K.CD, and 
38 this resolves itself into 
) solution e and a mixture 
Fiaure 90. of the two solids, namely, 
) ethylate and that with 
the composition of the point K(KOH. 2H,O). The action 
is simply hydrolysis; the aleohol produced forms the 
solution with some of the caustic potash and some of the 
water, while the rest of the potash and the water form the 
solid A. For a particular complex m the proportions of 
solution and respective solids may at once be read from 
the diagram in the way frequently illustrated in the pre- 
ceding pages. 
Further addition of water leads to a steady decrease of 
the ethylate, until at ~ it has all vanished and of the 











’ 
— SL 
’ 


Three-Component Systems 143 


total complex the fraction ~A/eK is solution ¢ and en/eK 
is solid K. With more water the complexes pass within 
the triangle ecK from ~ to 0;,the solid phase remains the 
same, K, while the solution changes continuously from 
etoc. Beyond o, in the triangle clK, the same solid is 
present, but there are now two liquid phases in equilibrium, 
namely, those of compositions c and 4 respectively. The 
proportions between the three phases depend only upon 
the position of the complex along op, and are readily 
ascertained. 

At p the solid phase K vanishes, and the complex 
consists of the two solutions only, being mainly 0: beyond 
p the compositions of the two layers change along cr and 
bq, respectively. At g the amount of 7 is zero and there- 
after, from gq to AB, the system is single-phase and un- 
saturated. 

The curve cad is binodal, having a critical point at a 
where the two layers become identical in composition. 

The equilibrium systems given by complexes falling 
within the various fields may now be summarized. It will 
be assumed that the pressure has been arbitrarily fixed at 
a value making the system condensed, that is, without 
a vapour phase. This means that one degree of freedom 
has been exercised. 

Field BC.AB.abacef: one liquid phase: tervariant 
(isothermal bivariant). 

Fields CD. ef, Kec, Kba: one liquid and one solid phase : 
bivariant (isothermal univariant). 

Field cad: two liquid phases: ditto. 

» CD.eK: two solid and one liquid phase: univariant 
(isothermal invariant). 

In the field CD.K.AD three solid phases can exist, but 

no liquid. 


144 Three-Component Systems 


Of other possible three-component systems it is proposed 
to discuss only one general type, namely, that in which 
mixed crystals may be formed by components. For reasons 
that will there appear, this discussion is postponed to the 
latter part of Chapter VIII. 


CHAPTER VI 


FOUR-COMPONENT SYSTEMS 


General. It is not proposed to aim at anything ap- 
proaching a complete treatment even of condensed four- 
component systems. The possible complexity is naturally 
much greater than in the lower systems, and the working 
out of general types on lines similar to those there adopted 
is still incomplete: nor indeed is much to be gained by 
what becomes rather mechanical procedure unless specific 
experimental work leads to it. 

With four components and no arbitrary limitations upon 
any variables, the maximum number of phases which may 
coexist becomes six, and there are many possibilities as to the 
states in which the six may be present. There can be only 
one vapour phase, but there are no limitations theoretically 
to the grouping of the remaining five as liquids or solids or 
both. Such a system is invariant. Five phases give a uni- 
variant system : four a bivariant system : and soon. Where 
a degree of freedom is exercised in a choice of pressure or 
temperature, the variance of the system is correspondingly 
reduced. This will be so in all cases touched upon here. 
Thus, as in the previous chapter, the vapour phase will not 
be considered ; that is to say, systems will be regarded as 
condensed, or subjected to pressures greater than those at 


Four-Component Systems 145 


which vapour phases can exist. If in practice one does 
not trouble actually to maintain this pressure, but works 
at any convenient value such as that of the atmosphere, 
the resulting changes in any system will be, as already 
pointed out in the similar cases of ternary systems, within 
the limit of ordinary experimental error. In condensed 
quaternary systems a single phase will be quadrivariant ; 
two phases, tervariant; three, bivariant ; four, univariant’; 
five, invariant. 

Graphical Representation. Increase of components means 
increase in the difficulty of adequately representing 
systems graphically. It has been seen that three inde- 
pendently variable components may be plotted in two 
dimensions either by means of a triangle or a square. 
A fourth component can be included only in a three- 
dimensional figure, so that the isobaric isotherm in four- 
component systems will be a space model, and one may 
no longer plot composition and temperature in one figure 
unless arbitrary limitations be imposed regarding at least 
one component. 

The tetrahedral representation of composition. Any four 
components 4, Bb, C, D, will, when each one of them in 
turn vanishes in amount, give the four three-component 
systems, B-C-D, A—C-D, d—b-D and A-B-C, In view 
of the usefulness of the equilateral triangle in representing 
the latter, it is natural to suggest four triangles, arranged 
to form a regular tetrahedron, as suitable for plotting 
isotherms of four components. 

In Figure 91, showing a tetrahedron in perspective, the 
angles 4, B, C and D represent respective pure components. 
The six possible binary systems, 4-B, A-C, A—D, B-C, B-D 
and C—D, are given by the lines joining the angular points. 
The ternary systems will be in the four bounding triangular 


2568 K 


146 Four-Component Systems 


surfaces, while one is able to represent every possible qua- 
ternary mixture by points in the space enclosed by these. 
As before, each side of a triangle may be taken as equal 
to 100 units, and compositions may be expressed in per- 
centages either by weight or by molecule. If now, 
through any point 
mS P, taken’ within 
the tetrahedron, 
parallels be drawn 
to the respective 
edges, meeting the 
bounding surfaces, 
four regular tetra-— 
hedra will be formed, 
as indicated in 
Figure 91, with P 
as common apex. 
The lengths of edge 
in these tetrahedra 
are Pa, Pb, Pe and 
Pd, respectively. 
The simple construction of the dotted lines makes it obvious 
that 











FIGURE 91. 


Pat+ Pb4+ Pe+ Pd = aat+al’+ac’ +ad’ 
=c'd'+ab’=CD; 

or, the sum of these distances is equal to the length of 
edge of the large tetrahedron, 100 units. If then Pa 
( = aa) be proportional to the percentage of component J, 
Pb (= 46) to that of B, Pe (= ce) to that of C, and Pd 

= dd) to that of D, this quaternary mixture is definitely 
represented by the point & and according to the position 
of such a point, every possible mixture of the components 
may be similarly represented. 


Four-Component Systems 147 


Applications. This method of plotting may be illustrated 
by applying it to two or three simple general types, 
selected according to whether a component be classifiable 
as a solid solute or as asolvent. Following Schreinemakers, 
the cases chosen will be those where the components are: 

I. One solid, 4, and three liquids, B, C and D. 

II. Two solids, A and C, and two liquids, B and D. 
III. Three solids, 4, B and C, and one liquid, D. 
Further, for the sake’ 

of simplicity, it will be 
assumed that only one 
liquid phase may exist 
in each case. 

I. One solid: three 
liquids, Those isotherms 
for case 1 which hold 
for temperatures below 
the melting-point of 
pure dA and above the 
freezing-points of Lb, C 
and D will take the a 
form of Figure 92. or as 

The points 4, c and d on the edges 44, AC and AD 
represent, respectively, the solubilities of 4 in B,C and D. 
The ternary saturation curves(compare Figure 42, Chapter V) 
are bc, cd and 6d. ‘The quaternary saturation surface is dcd. 

In every case two phases are present, the system being 
regarded as condensed, that is, the pressure so chosen that 
the vapour phase does not appear. ‘There are therefore 
three degrees of freedom possessed by every two-phase 
complex in equilibrium: temperature and the concen- 
trations of two components may be varied without either 
phase disappearing or a new one appearing. 

K 2 


A 





148 Four-Component Systems 


In the space BCDbcd all mixtures are unsaturated 
liquids, single phases. Single liquid phases of com- 
positions in Ated would be supersaturated and separate 
into solid A and a liquid on the surface dcd. Thus the 
complex s would give the proportion s¢/4¢ as solid 4 and 
As/At as liquid of the composition ¢. The number of 
phase regions into which the tetrahedron is divided in 
this very simple case is thus two. 

Il. Two. solids: two 
liquids, For ease II, 
where A and C are 
solids and B and D 
are liquids, the iso- 
therm takes the form 
of Figure 93. 

Two of the ternary 
systems, namely, ABD 
and CBD, are of the 
same kind as were all 
four in Figure 92, each 
giving a single curve 
(UV or bd’) of solu- 
tions in equilibrium with a solid (4 or C). In the remain- 
ing two, ABC and ACD, where in each case there are two solid 
solutes and one liquid, two intersecting curves are given, 
similar in type to the right-hand portion of Figure 44 
in Chapter V. Thus along 4/4 the solutions are in equi- 
librium with solid 4; along 46 with solid C. At d the 
solution and the two solids, 4 and C, coexist. Similarly 
with solution d the same two solids are in equilibrium. 
In their respective ternary systems, 4 and d are (con- 
densed) univariant points: alteration of temperature will 
not reduce or increase the number of phases. In the 


A 





FIGURE 93. 


Four-Compo nent Systems 149 


quaternary system they are bivariant points. Temperature 
being fixed they become univariant;. the composition 
of the liquid phase may then be altered by addition of 
another component and one obtains a curve 4d giving 
quaternary solutions in equilibrium with solid A and 
solid C. 

There are now two saturation surfaces; points on the 
one, U’bdd’, are in equilibrium with 4; on the other, 
b’bdd’’, with C. The space behind these surfaces includes, 
as single phases, all unsaturated solutions. The space 
in front includes mixtures which will separate into two or 
more phases, and it consists of three distinct portions. Of 
these, the first is bounded by the five surfaces 40’d’, Al’d, 
Abd, Ad’d and bdd’b ; all complexes represented within it 
will give solid 4 and a solution on the last-mentioned 
surface. ‘The second is similar. It is bounded by Cid”, 
Cb"b, Chd, Cdd” and bdb’d’’ ; complexes within it resolve 
themselves into solid C and a solution on this last surface. 
The third is different. It includes the remainder of the 
upper space and is bounded by the four triangular surfaces 
ACb, ACd, Abd and Cid. Complexes falling within it give 
two solid phases, 4 and C, and a solution lying on the 
line dd. 

III. Lhree solids: one liquid. The third case, where 4, 
B and C are solids and D is a liquid solvent, gives rise to 
isotherms such as Figure 94. 

The resemblance to Figure 41, Chapter V, is quite 
marked. In that case formation of liquid solutions from 
three solids was brought about by rise of temperature: in’ 
this case it is effected by addition of a fourth (liquid) com- 
ponent in which all three solids are assumed to be soluble. 
a, 6 and ¢ give the respective solubilities of 4, B and C 
in D. In each ternary system there are two curves meet- 


150 Four-Component Systems 


ing at a point where two solids and a solution coexist. 
The pairs of curves are: 

ad and bd in the system 4—b-D ; 

be and ec in the system b—C-D; 

cf and af in the system 4—C-D. 

In the quaternary system are three curves, dg, eg and fy, 
along each of which a pair of solid components and a solu- 
tion are in equilibrium. These three curves meet at the 
point g, where there 
is a solution satu- 
rated with 4, B and 
C. This will be 
an (isothermal, con- 
densed) invariant 
system. The posi- 
tion of g will vary 
with temperature, 
but at a_ selected 
temperature the 
three solids cannot 
be in equilibrium 
with any other solu- 


D 





FIgurE 94. 


tion than that of composition g. 

There are, thus, at the selected temperature and pressure 
of Figure 94, seven invariant saturation points of which 
one (g) is four-phase quaternary, three (d, e,f) are three- 
phase ternary, and three (a, 4, c) are two-phase binary. Also, 
there are nine univariant saturation curves of which three 
(dg, eg, fg) ave three-phase quaternary and the remaining 
six (ad, bd, be, ce, cf, af) are two-phase ternary: and there 
are three bivariant saturation surfaces (bdge, cegf and afgd), 
all of which are two-phase quaternary. 

In the space lying above these surfaces and extending to 


Four-Component Systems 151 


pure solvent D, only a single phase, unsaturated liquid, can 
exist. In it, therefore, the system is tervariant. 


D 








FIGURE 95. 


The portion of the tetrahedron lying below these satura- 
tion surfaces is polyphase, and there are seven distinct 
sections in it. The boundaries of these various regions 


152 Four-Component Systems 


may be followed from Figure 94, while in Figure 95 the 
several portions of the tetrahedron are represented separately, 
the lettering showing quite clearly how the parts fit together 
to form the whole model. 

System No. I, tervariant, single-phase, requires no 
further description. No. II is bivariant, the two phases 
being solid 4 and a solution lying on the surface adgf. 
Its bounding faces, other than the saturation surface, will: 
be traced out by a straight rod (or line) of adjustable © 
length with one end pivoted on 4 while the other travels 
round the edges of this saturation surface. III and IV 
are similar. V is univariant, the three phases being solid 
A, solid B and a solution lying somewhere on the curve dg. 
This region may be described as ‘double-gabled’, the 
gables being dg and 44, and the bounding surfaces being 
the three triangular planes ddg, 48g and Bdg. VI and 
VII are similar. Region VIII is invariant, or four-phase, 
three solids 4, B and C coexisting with solution of com- 
position gy. g is its apex, the triangle ABC is its base, and 
its other three walls are A4hg, Cg and ACg. 

It is on these various regions that attention should be 
focused, It is a mistake to attend too exclusively to the 
saturation surfaces, curves and points. These are boundaries 
- and important as such, but the essential matters are the 
regions which they bound. 

Projections of the tetrahedron. In simple cases, such as 
these three, a perspective drawing of the tetrahedron is 
readily made and serves to show clearly the different 
divisions. ‘To make it more than qualitative, however, is 
difficult : accurate plotting in a perspective drawing is not 
easy. It is therefore preferable in practice to work 
with projections on which plotting may be carried out 
readily and accurately. If two projections in different 


Four-Component Systems 153 


directions be given, the space model may always be 
reconstructed. 

Of the possible projections, those made orthogonally, 
that is to say, at right angles, to selected planes are by 
far the most useful. A perspective projection, on the other 
hand, may- give almost nothing at all, as for example that 
- from one of the angles of the tetrahedron to the opposite 
triangular surface in a case like that of Figure 92: while in 
such a case as Figure 94 it will give from D only the three 
curves @’g’, ¢y’ and 
Ig shown dotted in BY 
the triangle ABC. 

Two orthogonal 
projections are com- 
monly used. The 
first is at right angles 
to any of the four 
three-sided _ planes 
bounding the tetra- 
hedron. The second 
is at right angles to 
a plane drawn parallel 
to two sides of the tetrahedron which do not intersect one 
another. In Figure 96 is given as an example a projection 
of Figure 93 by the first method on to the plane BCD, 

The lettering is the same as in Figure 93, and a com- 
parison with it will make the relations clear without further 
words. 

The second method gives Figure 97 for the projection of 
Figure 98 orthogonally on to a plane parallel to both AC 
and BD, two edges which do not intersect. 

The six edges of the tetrahedron give four sides of a 
square and two diagonals. In the figure the latter are AC 








Fiaure 96. 


154 Four-Component Systems 


and, BD and their lengths are the same as in the model 
projected. The similarity to the original figure is greater 
in the second projection than in the first, and consequently. 








FIGuRE 98. 


it is usually preferred in 
practice. 

To plot the projection 
p’ of any point p is a 
simple matter. If p con- 
tains a units of A, 6 of B, 
c of C, and d of D, so 
that a+b+c+d= 100 
units = BD or AC, and if 
we regard BD and AC as 
axes X and J, respectively, 
and take o as the point 
of origin so that oD and 
oA (each 50 units) are in 
positive directions and 
oB and oC are in nega- 
tive, then for the point p’, 
yee d—b = eae, 
hence the point may be 
plotted when a, 4, ¢ and d 
are known. 

Formation of compounds 
between components, 'The 
studentof four-component 
systems should now pro- 








ceed to introduce into Figures 92, 93 and 94, or their 
projections, the necessary modifications which follow when 
one or more compounds may be formed between com- 
ponents. It is not proposed to do more here than show, 


Four-Component Systems 155 


in illustration, the alteration that follows in the simple case 
of Figure 92 if the solid 4 be able to form a compound (8) 
with the liquid B. The projection is given in Figure 98, 
where s gives the composition of the compound 4,8, 

The ternary system ABC has now two curves: Je gives 
solutions in equilibrium with solid S, and ec those with 
solid 4A. Similarly, in the system AD, there are two 


A (Li,S0,) 





C(WH,), S04] 
FIGURE 99. 


corresponding curves, 4f and fd. In ACD there’ remains 
as before only the one curve cd. 

Before the stage of this isotherm is reached, the region 
of existence of the compound may, of course, be confined to 
an area such as that between the dotted curve gf and the 
side AB. 

— Appearance of two liquid layers. When two liquid phases 
appear the number of quaternary surfaces may be largely 
increased. Again one case by way of illustration must 
suffice. Figure 99 represents a perspective view of a tetra- 


156 Four-Component Systems 


hedral isotherm for a system of components of which B 
and D are liquids (solvents) and 4 and C are solids. It is 
based somewhat upon one of Schreinemakers’s * isotherms 
for the system lithium sulphate (A)-aleohol (4)-ammo- 
nium sulphate (C)-water (1), omitting the hydrate of 4 and 
exaggerating the solubilities of the salts in B. 

The ternary systems 4-5-C and dA-D-C are of the type 
shown in the left-hand part of Figure 49, Chapter V. 
ab and mu, in the respective systems, give solutions in 
equilibrium with solid 4; d¢ and m/ those with a compound 
of d and C, say 4,C,; cd and /k those with solid C. The 
third system 4-5-D is of the simplest type with a single 

‘curve an of solutions coexisting with solid 4; while the 
fourth ternary system, L-C—D, shows two liquid layers and is 
of the type already represented in Figure 63 of Chapter V. 
de and gk are both curves of solutions saturated with solid C, 
the liquids e and g being in equilibrium with one another. 
The binodal curve is esg, and s is its critical point. 

The quaternary saturation surfaces, curves and points are 
apparent in the figure. All quaternary solutions lying on 
abgomn will coexist with solid 4: all on dc/thlmopgé with 
the compound of 4 and C: all on either area cdef or hgkil 
with solid C. fesght_is a binodal surface, and a liquid of 
composition corresponding with any point in Z, will be in 
equilibrium with some other liquid represented by a point 
in L,. The tie-lines between such liquid layers are not 
shown in the figure; but the points at which the two 
layers become identical lie along a critical curve st. 
Finally, there is the surface gropq, similarly binodal, with 
a critical curve 7p. This area lies wholly within the 
tetrahedron, no point on it being on any of the bounding 
ternary systems. ‘Two-phase equilibria are characteristic 

* Zeit, phys. Chem., 1907, 59, 653. 


Four-Component Systenis 157 


of all these surfaces, which are, therefore, (isothermal, con- 
densed) bivariant. 

On the quaternary boundary lines common to any two 
fields, three phases will coexist, the systems being uni- 
variant. The details may best be given as a table: 


Curves. Coeaisting phases, 
' bg and om Solution, solid 4 and solid 4, B,,. 

gro Two solutions and solid A, the two 
solutions becoming: identical at 7. 

gpo Two solutions and solid 4,C,,, the solu- 
tions becoming: identical at p. 

cf and h/ Solution, solid A,C, and solid C. 

Sth Two solutions (becoming: identical at /) 
and solid 4,C,. 

Je and gh Two solutions and solid C. 


Lastly, the solution represented at the quaternary point 0 
may exist in equilibrium with that at gand with the two’ 
solids A and 4,B,. Similarly for solutions at f and / 
and the solids C and 4,8,. Such four-phase systems are 
invariant when pressure and temperature are fixed. 

All points lying underneath these surfaces represent 
unsaturated solutions. The resolution of the space above 
them into divisions of which the various surfaces and lines 
are boundaries, will offer to the student an exercise the 
solution of which may be based on the same lines as those 
adopted above in the very much simpler case of Figure 94. 

Salt double decomposition in a solvent. The majority of 
the experimental investigations on four-component systems, 

~which have so far been carried out, have dealt with systems 
in which one component has been water and the others 
salts. These belong to the class of most importance in the 
technical branch generally termed ‘salt chemistry’. It is, 


158 Four-Component Systems 


indeed, almost solely in this section of technical work that 
the value of the study of heterogeneous equilibria from the 
systematic standpoint of the phase rule has been realized 
and its results applied in the devising of processes : though 
there is little doubt that in the separations involved in the 
manufacture of organic products as well, proper scientific 
procedure must be based on similar studies. 

Certain salt equilibria may differ somewhat from other 
cases in consequence of the possibilities resulting from 
double decomposition or exchange of radicals. It has 
already been seen, at the end of the previous chapter, that 
in three-component systems where reaction of the kind is 
possible, the triangle is no longer suitable for graphical 
representation. So in the present systems, the regular 
tetrahedron ceases to be satisfactory when it is necessary 
to plot the relations between four components, three of 
which are connected by the equation 

AB+CD— AD+CB. 

For if 4b, CD, AD and solvent be the components 
represented at the angles of the tetrahedron, it 1s not 
possible to represent a composition C4 within the bounds 
of the figure. But one may deduce from the tetra- 
hedral method a satisfactory representation which at once 
recalls Figure 89 in Chapter V. The deduction is instruc- 
tive and worth following. 

Deduction of method of representation. ake first the 
ternary system 45+ CD) — AD+CB, and represent each 
phase in terms of the radicals 4’, B’, C, D’, the amounts 
in any particular phase being w, #2, y and ¢ equivalents 
respectively. Assume for the moment that these radicals 
are independent variables and form a quaternary system. 
They may then be represented by the respective angles of 
a tetrahedron. If this tetrahedron 4’ B’C L” be projected 


Four-Component Systems 159 


on the plane parallel to 4C and LD one obtains the square 
A b’C D’ of Figure 100, the four sides and two diagonals 
being the projections of the six edges of the tetrahedron. 
If now the length of each side of the tetrahedron, that 
is, w+@+y+2, be taken equal to 2 (or 200), then in the 
projection the diagonals 4°C" and B’L’ = 2 (or 200), and if 
they intersect at 0, OA” = OB’ = OC’ = OD’ = 1 (or 100). 
The four salts 
AB, BC, CD and 
AD are then 
compounds of the 
components and 
lie at the middles 
of the sides of 
the square 
ABCT. 
The point p will 
now _ represent 





the phase 

PLES ORES oe ‘ a : 
if, in the con- pape ee eer Ge ae pee Ay 
struction set out Figure 100. 


in Figure 100, 
pg = andaeg ei aia | 

A consideration of Figure 91 and its relation to the 
orthogonal projection here used will at once show this 
to be the case, as has also been pointed out in discuss- 
ing Figure 97. Similarly, all possible combinations of 
A’, B’, C and D’, considered as independent variables and 
without restrictions consequent upon their positive and 
negative characters, may be represented by points in the 
square 4 B’C" D’.- 


160 Four-Component Systems 


If, however, there be a relation between the components 
sach as exists when two are positive and two are nega- 
tive radicals, as in a reciprocal salt pair, and therefore 
w+y=2+2=1, it follows that all possible phases must lie 
in the square 4B. BC.CD.AD; for complexes lying without 
it will not contain the necessary equality of positive and 
negative radicals. Thus a point in the triangle 4B. BC. B’ 
represents an impossible mixture of two salts dB and BC 
with uncombined radical B’. Moreover, this relation 
between radicals reduces the number of mage cee com- 
‘ponents to three. 

Drawing the diagonals BC. AD and AB. CD, one obtains 
four triangles, 4B.BC.AD, BC.AD.CD,CD.BC.AB and 
AB.AD.CD, and it is a simple geometrical: exercise to 
show that p represents the same composition whether 
considered as a point in one of these triangles or as a point 
in the square 4. B’C D’. For example, in the triangle 
AB. BC.CD, p represents 

pa equiv. of CD + pk of BC + pil of AB, 
where pat+pk+pl=AB. BC=$AC =1. 

This means that w=pl, e=pk+pl, y=pa+t pk and z=pa, 
a—z pk+pl—pa _1—2pa 





whence 


a 
_ k—pl 1—2ypl 
and " 50 _ oe —_ —- =e 


and these are the conditions that p should represent 
A,,B,C,D, in the projection square A’ B’C’D’. So for the 
aoe fianele 

Next consider the following aunplinceiem First note 
that pe= i(l—pa—p)\ V2) ee) 
and pf = 1(pb—pa\/ 2 
Expressing p in terms of 4B, BC and AD, we have present 

pa equiv. of AD +pb of BC+ (1—ap—Op) of AB, 


Four-Component Systems 161 


or if a phase of this composition contain all the four possible 
salts of the equilibrium 48+ CD — BC+ AD, there will be 
(ap—n) equiv. of AD, (pb—xn) of BC, (1—ap—bp +x) of AB, 
and x of CD, Putting 

1—ap—b = equiv. AB—equiv. CD 


and pb—pa = equiv. BC— equiv. AD, 
it follows from (1) and-(2) that 

pe = £ (equiv. AB—equiv.CD)/2  . . (8) 
and pf = k (equiv. BC—equiv. AD)/2  . . (A) 


But AC = B’)’=2; therefore each half - diagonal 
0.AB, O. BC, O.CD and 0. AD must equal 1/2. 
If, then, one chooses to put O.AB=0.BC=&.=1 
instead of 44/2, equations (8) and (4) change to 
pe = equiv. 4B—equiv. CD 
and Df = equiv. BC— equiv. AD, 
and one obtains the following simplified method of graphi- 
cally representing the reciprocal salt pair system : 

Take two axes bisecting one another at right angles at O 
and each equal to length 2. Taking directions O. BC and 
- O.CD (Figure 91) as positive, and O.AB and O.AD as 
negative, plot along the one axis from QO, as origin, the 
difference in equivalent proportions of BC and AD and 

along the other of 46 and CD, the sum of these four 
molecular proportions being unity: or, briefly, taking 
bisecting axes each = 200, plot the differences in _per- 
centages of equivalents of the respective pairs. 

We thus have in Figure 100 three different methods of 
plotting the composition of a phase, each method giving 
the same point y. ‘The third method is the one which will 
be used. To represent the fourth component water, or 
other solvent, it is now only necessary to take a new 
axis OW at right angles to the square at O. Putting 
OW=0.AB=Kce.=100, and plotting the molecular per- 


2568 L 


162 Four-Component Systems 


centage of water upwards, VW becomes the point represent- 
ing pure water, and all mixtures of the four components 
must lie in the pyramid W.AB.bC.CD. AD, 

As two-dimensional projection, that made orthogonally 
to the ground plane 4B. BC.CD.AD may be taken, and 
from this and a second projection, say that on the plane 
W.AB.CD, the solid pyramidal diagram may be recon- 
structed. It goes without saying that from &. projection 
one cannot read directly the compositions of phases. For 
example, the point O in Figure 100 represents either pure 
water or any solution whatever in which BC and AD (or 
AB and CYP) are present in equivalent proportions. 

Example of pyramidal model. The simplest possible 
reciprocal salt pair system is that in which neither 
hydrates nor double salts are formed. Figure 101 repre- 
sents in perspective the space model of a possible system of 
this kind at a particular temperature, and its orthogonal 
projection on the base of the pyramid, while this projection 
is shown in direct view in Figure 102. 

The description of Figure 101 (and its projection) offers 
little that is new, but the case is so important that it is 
advisable to go briefly into the dissection of the pyramid. 
If the case of Figure 94 has been followed, that of Figure 
101 will not offer difficulty. 

The edges of the pyramid and the diagonals of the base 
give all the possible binary systems, ten in all. We are 
concerned only with the four in which, at the temperature 
of the figure, a liquid phase can appear, namely, those 
running from the apex (water) to each corner of the base. 
a,c, e and g give saturated solutions of the respective 
single salts in water, condensed isothermal binary systems. 
The four triangles having water at their common apex 
contain the ternary systems, each assumed to be of the 


Fowr-Component Systems 163 
type of the right-hand portion of Figure 44, Chapter V. 
6,h, f and d are the three-phase ternary systems, invariant 
under the conditions laid down. 


The entry of a third salt to each of the ternary systems 
adds a degree of freedom and brings all solutions to the 





‘ = Ze 
4 =sS 
pea Se ae 
/ -——\ 


ee 
SO Za 


2eN Nae ih (i) ol os eee pececneeceeene ee enceceee neste BC 
Pe a’ b Cc’ 
AD . CD 
Figure 101. 





interior of the pyramid: quaternary saturation curves run 
from each of the three-phase ternary points. 

b and / are represented as meeting at 4, where therefore 
four phases (solution and the three salts 4D, 46 and BC) 
coexist, a quaternary (isothermal, condensed) invariant 


point. Those from d and / meet at /, a solution saturated 
simultaneously with respect to BC, CD and AD. Any 


ie 


Those from 


164. Four-Component Systems 


attempt to alter the composition of #4 or /, by dissolving 
CD or AB as the case may be, will result in one of the 
solid phases disappearing, namely, 48 from the system at 
& and CD from that at /. There is therefore another 
saturation curve running between / and / giving solutions 
saturated with respect to two solids only, BC and AD. 

These various curves are the boundaries of four satura- 
tion surfaces. Any point on the surface hadk/ represents 
the composition of a solution saturated with respect to 4B; 
on ghkifg with respect to 4D; on eflde to CD; on cbhldec 
to BC. With these known, it is possible at once to divide — 
the pyramid into sections according to the nature and 
number of the phases into which complexes of mean 
composition falling within such sections will resolve 
themselves. 

There are twelve of these sections. The first and most 
obvious one is that lying between the apex and the four 
saturation surfaces. All mixtures in this give unsaturated 
solutions. Such single-phase systems are tervariant when 
two degrees of freedom have been exercised in fixing 
temperature and pressure. Concentrations of three com- 
ponents may be varied, within limits, without a new phase 
appearing. The remaining eleven are below these surfaces 
and are of three types. Of the first type there are four, of 
the second five, of the third two. 

An example of the first type is that region which is 
bounded by the area habs and the trace of a straight rod 
(or line) of adjustable length moving with one end on the 
angle AB as a pivot while the other end passes round the 
boundary of the area just cited. Any complex in this 
. resolves itself into two phases, solid 4B and a solution on 
the saturation surface. The system is (isothermal, con- 
densed) bivariant. The other three of this type may be 


Four-Component Systems 165 


similarly described with respect to the remaining saturation 
surfaces and their corresponding solids. 

To the second type, the term ‘ double-gabled ’ may again 
be applied. An example is the region in which 4/4 and 
AD, AB ave the gables, and the bounding surfaces are the 
four triangles 4,4 D.AB, k.AD.AB, h.AD.& and h. AB.&. 
Complexes in it yield two solids, 4D and AB, and a 
solution of composition somewhere on Af, saturated with 
both. The condensed systems are univariant at a fixed 
temperature. The concentration of one component in the 
system may be varied, within limits, without altering the 
phase equilibrium. The other divisions of this type have, 
as their pairs of gables, 04 and AB. BC, di and CD. BC, 
f/and AD.CD, and #/ and AD. BC, respectively. 

Two examples of the third type divide the remainder of 
the pyramid. One has an apex at 4 and its base is the 
triangle 46.AD.BC: its sides are the three triangles 
hk. AB. AD, k. AB.BC and &.4D.BC. The other has / as 
apex, 4D.BC.CD as triangular base, and corresponding 
triangular sides. Complexes in either of these give a 
solution, / or /, as the case may be, and three solids. They 
are (isothermal, condensed) invariant systems. 

It is clear now that one can predict from this three- 
dimensional isotherm precisely into what phases any pos- 
sible mixture of components will resolve itself. At other 
temperatures the division of the model between the various 
systems of phases may be quite different. That which lies 
in one type of system at one temperature may le in an 
entirely different one at another. From its equilibrium 
with a solution of given composition it may be possible to 
remove a certain component at one temperature, while 
below or above this quite another may separate out. Nor 
is variation of temperature the only method by which one 


166 Four-Component Systems 


_may pass from one system to another; obviously this may 
be done by suitable additions, or removals, of, say, the com- 
ponent water; or by both means together. It is in this 
way that one may use these systematic investigations for 
the devising of technical processes for the production of 
solid salts (4B and CD, say) by double decomposition 
between two others (AC and 4) in presence of a solvent. 

Looking now at Figure 102, the projection of 














Fieure 102. Figure 103. 


Figure 101, it is quite apparent that there may be another 
relative division of the plane, namely, that shown in 
Figure 103. 

In this case, before the curves from 4 and / meet at 4, 
a new phase, solid CD, separates out at # and /’, respec- 
tively. The difference between the system of Figure 102 
and this one is that in the former BC and 4D can coexist 
with a series of solutions (along 4/), while 4B and CD 
_ cannot coexist as solids with any solution, whereas, in 
Figure 108, 4B and CD can, while BC and AD cannot, 
coexist with solutions. 


Four-Component Systems 167 


The (condensed) invariant point. Now whether the one 
case or the other will be realized in practice will depend 
first of all upon the natures of the salts, but secondly upon 
the variables temperature and pressure, or what is not 
shown in the projection, water content. The effect of 
pressure is usually very slight, at any rate within the 
limits of pressures so far realized experimentally, but tem- 
perature may modify these diagrams considerably, so that 
the isotherms above 
a certain point may BC 
be of the one type, 
and below it of the 
other. At that par- 
ticular point 4 and 
Zor # and VU co- 
incide, and the rela- 
tions are those of 
Figure 104. 

This point of 
coincidence (K) is 
a five-phase (four : a 
solids and solution), Preven 104. 
or condensed in- 
variant, system. It can exist only at one specific value for 
temperature and for each of the concentration variables, so 
long as the selected constant pressure is maintained. One 
phase or another disappears if a persistent attempt be made 
to alter one of these. 

Course of isothermal evaporation. It is a simple matter to 
follow out from a model like Figure 101 the course of 
isothermal evaporation of an unsaturated solution. Thus 
a quaternary solution 2 will, on loss of water, travel along 
the line H,O.# produced until it meets one of the surfaces 











168 Four-Component Systems 


at, say, y. Here solid (BC) will begin to separate. With 
further evaporation and formation of BC, the solution com- 
position moves across this surface, alone what may be 
termed an evaporation line, until a boundary surface dd, 
say, is reached, when a second solid (CD) begins to deposit 
as well: further change would then be along this line to 
a point (/) where three solids (BC, CD and AB) would 






CuCl, 2H,0 








(NH,C1), 
Figure 105. 


separate. The solution would finally vanish at this point, 
which is sometimes termed a ‘ drying-up point’. 

Example in which components form compounds. When 
hydrates (or in general, solvates) and double salts may be 
formed, the model naturally becomes more complicated, but 
the complication consists only in greater numbers of the 
systems illustrated above and not in the addition of any- 
thing new. The projection of rather an interesting example 
is reproduced in Figure 105. The reciprocal salt pair is 


CuSO,—(NH,Cl),— (NH,),SO,—CuCl, and the solvent is 


Four-Component Systems } 169 


water. The relations are for 30° C., and the figure is from 
a paper by Schreinemakers (Zeit. phys. Chem., 1909, 69, 
557). 

- The significance of the various surfaces, curves and points 

should be at once apparent with the aid of the formulae 
entered on the figure. Instead of the four isothermal, con- 
densed, bivariant, quaternary areas of Figure 101, there are 
here six: instead of the two invariant points / and /, there 
are four, namely, p, 0, m and /. It will be noticed that 
neither copper sulphate pentahydrate and ammonium 
chloride, on the one hand, nor copper chloride hydrate and 
ammonium sulphate, on the other, can exist in equilibrium 
with one another and a common solution. 
_.A point worth attention’ is the relation between water 
and the double salts, CuSO,.(NH,),SO,.6H,O and 
CuCl, .2NH,Cl.2H,0. The two parts, sO and Of, of the 
straight line s¢ give the projection of the lines joining the 
compositions of these salts, respectively, to the apex of the 
pyramid, which is pure water. It cuts cd, the curve along 
which solutions are in equilibrium with the first double 
salt, at r, and similarly gh at g. This means that each of 
these double salts dissolves in water without decomposition. 
Moreover it is clear that a ternary system is possible in 
which the components are water, CuSO,.(NH,),SO,.6H,O 
and CuCl,.2NH,Cl.2H,O. The isotherm will consist of 
two curves, one made up of solutions projected along rz, 
and the other of those projected along qu. At 2, where 
the curvature changes sharply, both double salts coexist 
with the same solution. 

Determination of solid phases. 'The determination of the 
solid phase in contact with any solution may be carried out 
by adopting precisely the same principle as that described 
in the chapter on ternary systems. The compositions of 


170 Four-Component Systems 


solution, wet residue (solid with adhering solution) and pure 
solid or mixture of solids, will lie on a straight line, and 
the application of this fact to determine the latter differs 
in this case only in that the line runs through a three- 
dimensional figure instead of on a plane surface. 

Common modifications of the method of representation. As 
many workers on reciprocal salt pairs prefer to modify 
slightly this pyramidal method of representation, it is 
advisable to mention the changes they introduce. Instead 
of expressing compositions as (equivalent) molecular per- 
centages of the respective four salts and the solvent, they — 
express them in terms of the numbers of molecules of the 
salts present with a fixed number, usually 1,000, of molecules 
of solvent. The differences between the molecular propor- 
tions of BC and AD are, as before, plotted against those of 
AB and CD, and then vertically to each such point is 
plotted the total sum of the salt molecules of all kinds 
present. Any point so obtained is of course quite un- 
ambiguous. | 

Further, for convenience, it is usual to decide arbitrarily 
that BC and AD, or AB and CD, shall be regarded as not 
occurring together in solution, or as being ‘ incompatible ’ 
as solutes, and therefore to tabulate all salt mixtures in any 
solution in terms of 48, BC and CD, or 4B, AD and CD 
(or in the corresponding sets of three if 48 and CD be 
selected). This does not affect the position of a point 
representing the composition of a phase. 

Some good examples of diagrams obtained in this way 
will be found in papers by Hillebrand* and Blasdale + 
recently published. In the space model built in accordance 
with this modified method, solutions will of course lie below 


* Journ. Indust. and Eng. Chem., 1918, 10, 96. 
+ Ibid., p. 344. 


Four-Component Systems iba 


the surfaces, lines and points of the saturated solutions, not 
above them as in the pyramidal model. Perhaps this 
method is a little simpler in some respects, but the advan- 
tage, if any, is slight in practice, and there is no advantage 
at all from a student’s point of view. 

Other workers, particularly Jinecke (Gesdttigte Salz- 
losungen, Kapitel X), make use of the mode of representation 
for three components shown in Figures 89 and 90 of 
Chapter V. It has been seen (p. 141) that this amounts to 
plotting the ratio of one of the positive radicals to the sum 
of the two against a similar ratio for the negative radicals. 
The fourth component is then plotted upwards either as 
a percentage or as the amount present for a given total 
number of molecules of salt. The method offers no particular 
advantage over the pyramidal, and it would-be well if the 
latter were adopted generally. 

Higher Systems. With the exception of the classic work 
of van’t Hoff, investigations on systems containing: more 
than four components have not very often been carried out 
to any degree of completeness. With a fifth component, 
the sum of the phases and degrees of freedom becomes 
seven, one more than in the quaternary systems. It is then 
no longer possible to represent all the phase relationships 
in a three-dimensional concentration figure unless some 
further limitation be introduced reducing the variability of 
the system. 

Van’t Hoff’s work dealt with aqueous solutions of salts, 
in particular with chlorides and sulphates of sodium, potas- 
sium and magnesium : in other words, with mixtures con- 
taining two negative and three positive radicals connected 
by the relation that the numbers of equivalents of each type 
of radical must be equal. Huis prime object was to study 
the conditions under which these salts, or compounds one 


172 Four-Component Systems 


with another, separate during natural evaporation of such 
waters as those from which the successive layers at Stassfurt 
were deposited. These waters (for example, ordinary sea 
or salt-lake water) on evaporation first deposit sodium 
chloride, so that the salts which appear later separate from 
a solution which is at the same time saturated with sodium 
chloride. If one, then, lays down the condition that solid 
sodium chloride shall always be one of the phases present, 
such a system of two negative and three positive radicals 
in water may be regarded as one of only four independent 
components. ‘The representation of isotherms by means of 
a space model then becomes possible. 

It is only by such means, or under such limitations, that 
one may graphically set out the results of solubility investi- 
gations in systems of more than four components. It is 
beyond the scope of this book to diseuss such salt work of 
the kind as has been carried out by van ’t Hoff, Meyerhoffer 
and several other brilliant investigators, Any one interested 
in the matter is advised to read, apart from the original 
memoirs, a survey of work done on the oceanic salt deposits 
which was published by Ernst Janecke in the Zeitschrift 
Siir anorganische Chemie for 1918. 


CHAPTER VII 


SOME THERMODYNAMICAL CONSIDERATIONS 


THE application of thermodynamics to equilibria of the 
types that have been considered can hardly be said to have 
led very far on the purely practical side. The generaliza- 
tions which are obtainable contain functions the quantitative 
evaluation of which is not possible owing to the unknown 


Some Thermodynamical Considerations 173 


factors involved exceeding in number the known inde- 
pendent equations connecting them. 

Apart, however, from the side of purely practical applica- 
tion, the generalizations are of high interest, and it is 
intended here to give briefly the chief of them and then to 
show in the next chapter how use may be made of certain 
of them from a qualitative standpoint. The discussion 
will, of necessity, be very slight, and students wishing to 
understand the subject thoroughly are advised to read 
Sackur’s Thermochemistry and Thermodynamics (Macmillan 
& Co.), Partington’s Thermodynamics (Constable & Co.), 
Planck’s Thermodynamik (von Veit und Co., Leipzig), or 
Lewis’s System of Physical Chemistry, vol. ii (Longmans & 
Co.). 

The First Law. The first law of thermodynamics, in 
accordance with which all changes in heterogeneous systems 
take place, is a special case of the principle of the conserva- 
tion of energy and is best expressed in the form 

dQ =dU+dW. ; 

Here dQ signifies a small amount of thermal energy 
absorbed by a system (though not the differential of a 
specific finite quantity @*) which results in increasing its 
internal energy by dU and effecting an amount of external 
work dW, usually by expansion against pressure. 

In any particular change the signs of these quantities 
may be positive or negative. It was supposed by Julius 
Thomsen (1851), Berthelot (1868) and others that in any 
change which occurred spontaneously without addition of 
energy from some external source, dQ must be-negative : 
that is to say, heat must be evolved. They regarded the 
heat of reaction as a measure of the force driving the 
system towards its equilibrium state. The equilibrium 

* See Planck, Thermodynamik, 4th edn., 1913, foot-note to p. 55. 


174 Some Thermodynamical Considerations 


conditions described in the foregoing chapters could be 
attained therefore only through exothermic actions. As 
a matter of fact, however, many quite common spontaneous 
chemical reactions are endothermic, or proceed with absorp- 
tion of heat, and the number is greater at higher tempera- 
tures than at lower. Some other quantity than heat evolved 
must therefore be the factor determining the direction of 
an action, and the first law of thermodynamics does not 
suffice to indicate it. 

The Second Law. From the second law, however, the 
conditions may be deduced under which a system will be in 
equilibrium either when completely isolated or, as is more 
usual in the cases which have been considered, when in 
some definitely prescribed relation to its surroundings. 

The second law, which like the first has an empirical 
basis, may be regarded as being concerned with the possi- 
bility of converting heat into work. While in accordance 
with the first law a definite relation exists between heat 
and work, and while work may always be converted com- 
pletely into heat, the reverse change of heat into work is 
never observed to be complete in any natural process. In 
other words, heat cannot, as a matter of experience, be 
converted into work as the one and only result of a spon- 
taneous process. Some other change always occurs as well, 
and this change is thermodynamically irreversible. The 
raction of the total heat supply which may be converted 
into work* depends upon the particular process adopted, 
but there is a definite limit to it which cannot in any 
circumstances be exceeded. This limit is attained in 
a process which is at every stage reversible, in the sense 

* This fraction is known as the ‘efficiency’ or the ‘conversion 


ratio’, alike of the process employed and of the machine which 
effects it. 


Some Thermodynamical Considerations 175 


in which the term ‘ reversible’ is understood in thermo- 
dynamics. 

Reversibility. A strictly reversible process in this sense 
is one which passes through a continuous series of equi- 
librium states. The driving force in such a case must be 
infinitesimally small, and this means that the velocity of 
the change will also be infinitesimal. In practice such 
processes are not realizable. In every spontaneously occur- 
ring natural process the driving force will be finite and the 
process therefore not completely reversible. The fraction 
of the heat of a given system which may be converted into 
work (and thence back into heat) will be a maximum, and 
the fraction spent irreversibly on other changes a minimum, 
in limiting cases to which one cannot in practice do more 
than approximate. 

The nature of the machine * used in this conversion is 
immaterial so long as the condition of reversibility is ob- 
served. Any such machine must work between specific 
temperature limits, say 7, and 7, on the absolute scale. 
A quantity of heat, Q,, will be taken in at 7;. A propor- 
tion of it will be converted into work, but it will be impos- 
sible to obtain work equivalent to a certain balance, Q,, 
which must be given up at 7), the lower limit of tempera- 
ture at which the machine works. 

From considerations of such cycles as that of Carnot it 
may be shown that 

Qi- Qe sand Zi tai 1 
: (eaeaT. siabeebla (ai)? 
which means that of the total heat Q, taken in, the maxi- 
mum fraction which may be converted into work is equal 
to the ratio of the temperature range over which the process 





* The term ‘machine’ includes two essential elements ; one is the 
working substance, the other is the mechanism by which it operates. 


176 Some Thermodynamical Considerations 


occurs to the temperature at which this quantity Q, was 
received. 

The complete conversion of heat to work would he possible, 
therefore, only in the impracticable case of a machine work- 
ing reversibly from 7', to absolute zero. In this case only 
does the efficiency attain the value unity. The maximum 
efficiency between 7', and any other temperature than zero 
is less than 1; and the relative efficiency of any process not 
completely reversible will be less than this maximum, 
that is / 

Wa Ga - ! ae Ma 
Q) 


+ 1 P 
Adequate discussion of these matters must be sought in 


the text-books cited, or others. 
Entropy. “quation (1) may be written in the form 
Qi eae 


Wh OST eee 

cd 
or if the heat absorbed, Q,, be considered as positive, and 
heat given out, Q,, as negative in sign, 


Q; Q. _ 
7 9h ee . py 


Where, however, the efficiency is less than the maximum 


possible, it follows from equation (2), introducing the same 
convention as to sign of Q, that 


@ , & eg | mee tt 








This holds for any simple eycle, that is to say, for any 
cycle in which the addition or subtraction of heat takes 
place at two temperatures only. But it may readily be 
shown that any given cycle at all, such as that represented 
in Figure 106, from state 4 to state B along the path 
ACB and then back again to 4 along another path BDA, is 
equivalent to a sum of simple cycles, and the greater the 


Some Thermodynamical Considerations Lhe 


number of the simple cycles the more closely does their 
course approximate to the given one. 

Hence in the limit, the equation (8), characteristic of 
a simple reversible transformation, becomes 


ee ont or ER Gy. 


where the integral on the left is made up of two inte- 
orals, 


B Zhe 
| aq along the path ACB, and | dQ 
hel Bee 


alone the path BDA, 
which are therefore equal B 
in magnitude and of oppo- 

site sign. C 
Since these integrals are 
taken along different paths, 
their common magnitude 
must be independent of 
these paths, depending only 
upon the terminal states 
A and B. It therefore Seco et aclon 
follows that either of these a ae 
integrals represents a fundamental difference between the 
respective values, characteristic of the states d and B, of 
some function of the variables by which these states are 
defined. Writing this function as S we have 


Intensity Factor 





a0 
S,—S, = | Tp? 
A B “ T 
or Tene Se ol hie alee are (a) F 


To this function S the name ‘entropy’ is given. In a 

change which proceeds at every stage with maximum 

efficiency of conversion of heat to work (i.e. without irre- 
2568 M 


178 Some Thermodynamical Considerations 


versible change), to which, therefore, equation (8) applies 
at every stage, we may write, for each such stage, 
= = 0, that is, dS = 0. 

Irreversible change. For every change, however, where 
the efficiency is not a maximum and where, therefore, 
equations (2) and (4) hold, 

a < 0. 
These are the kinds of change we find occurring spon- 
taneously in nature, in the processes of solution, vaporiza- 
tion, crystallization, condensation and the like. In these, 
the driving force is not infinitesimal, and reversibility is 
not realized. If the change be from state 4 to state B, 
it follows then that nature has, so to speak, some ‘ prefer- 
ence’ for state B, since the system passes to 6 spontaneously 
but cannot return from B to 4 without the expenditure of 
work upon it by some external agent. — 

Any quantity which gives a measure of this ‘ preference’ 
_ will enable us to predict the direction of change between 
any two states. This quantity must be the difference 
between the values of some function for the respective 
states, a function which increases in all irreversible changes 
and remains constant in reversible ones. Such a function 
must also, of course, depend only upon the variables 
defining that particular state and be quite independent 
of the previous history of the system before it attained 
that state. Moreover, the function must be single-valued, 
that is, any possible set of simultaneous values of the 
variables must specify only one value of the function. 

General condition for thermodynamic equilibrium. Berthe- 
lot’s assumption that heat of reaction was the required 
quantity was incorrect because the amount of heat given 


Some Thermodynamical Considerations 179 


out in changing from state 4 to state B is dependent 
upon the path taken by the process of change, that is, 
upon the complete history of the final system from its 
initial to its final state. When, for example, zine dis- 
places copper, the heat of reaction may amount to a 
considerable positive quantity: but if the displacement be 
carried out in such a way that an electric current is pro- 
duced, the residue of the energy change, which will appear 
as heat of reaction, may be negative. 

The function entropy, however, does comply with all 
the necessary conditions prescribed in the previous section. 
S,—Sp is the difference of two values of this function, the 
magnitudes of which are dependent only upon those sets 
of values of the variables which define the states 4 and 8B, 
respectively. 

From the equation 

§4—Sp = | ae 

Ped 2 eae Re 3 
which defines entropy change, it is obvious that, when the 
right-hand side is zero, Sy = Sp, that is, in a reversible 
change, entropy remains constant. When, however, the 
change is partially or wholly reversible the right-hand side 
is negative, so that S,<S,, or the change from d to 6 
results in an increase of entropy. This result is quite 
general for all isolated systems in which the changes which 
take place are entirely independent of any that may occur 
outside. 

Restatement of laws. To the first law, that the total 
energy of an isolated system remains constant whatever 
‘changes may occur in it, we may now add the second 
law, in the form that spontaneous change in such a 
system will occur only when the result is an increase in 





entropy. 
M2 


180 Some Thermodynamical Considerations 


Tnmitations in practical application. It is clear that these 
two statements might be of the utmost importance in 
the practical quantitative consideration of heterogeneous 
systems. Whether, for example, a mixture of, say, two 
simple salts and water will, or will not, spontaneously 
change into a hydrated double salt and a solution when 
no energy is introduced from without or given out by the 
system, depends simply upon whether the sum of the 
entropies of the components before reaction is less or 
greater than the sum of those of the resultant phases, and 
this in turn greater or less than that of any other arrange- 
ment. If therefore one could calculate entropy from a 
knowledge of the variables, pressure, temperature and 
volume composition for any and every phase, one could 
always predict the direction and extent of a change. 

Unfortunately it is not possible always to do this. For 
one thing, it follows from the definition of entropy as 

dQ 

gE 
that it contains an integration constant not determinable 
by thermodynamics which is, in part at least, characteristic 
of the substance or phase, but otherwise arbitrary. But 
quite apart from this, the integration as a function of the 
variables of condition is not possible unless an equation 
connecting: them is known. For a perfect gas such an 
equation is known, and we may readily calculate the entropy 
of, say, a gramme-molecule, in the following manner : 

In accordance with the first law, dQ =dU+dW. Hence 

dQ dU+dW 

ieee | 
Now the increase per degree per gramme-molecule of the 
internal energy of the gas, being independent (by the 
Joule-Gay Lussac Law) of the volume, may be written 








Some Thermodynamical Considerations 18] 


as CT, where C, denotes the molecular heat at constant 
volume. Also the work done, dW, may be written as pdv, 
hence 


4 d 
Seneaad oo 
or, since pv = RT’, ; 
aT’ dv 
ds = Ce 7 + R 5 ee o 
Integrating, S= C,loe7T+ Rlogv+S8’, 


where S’ includes a constant. characteristic of the gas, but 
is otherwise arbitrary. 

Where therefore one is concerned only with changes in 
the same gas, as in expansions and compressions, differences 
in entropy may be calculated in terms of measurable quanti- 
ties, #, v and 7’ (the integration constants disappearing) 
and the direction of change predicted. 

In the cases of liquids and solids, however, a general 
equation of condition is lacking, and we are unable from 
a knowledge of variables to predict whether, or in what 
direction, a given system will change. 

Conditions of equilibrium im systems under constraint. The 
law that a system is in equilibrium only when its entropy 
has a maximum value holds only if the system be isolated 
in the sense indicated above. But to effect such isolation 
is extremely difficult, and, in practical investigations of 
phase equilibria, systems are usually subjected to some 
condition, such as constancy of temperature or pressure, 
_ which can be maintained only by changes in bodies exterior 
to the system. In every such case the condition for equi- 
librium is different. Writing the first law, dQ = dU+dV, 
in the form 7dS = dU+dW, the equilibrium conditions in 
two cases of the kind may then be deduced in the follow- 


ing way: 


182 Some Thermodynamical Considerations 


I. Lsothermal isometric systems: free energy a minimum, 
When the condition of constraint is that the temperature 
be maintained constant, the simple system may be regarded 
as part of an extended system, which includes a thermostat 
of such large mass and inexpansible material that, when 
it gives up or gains heat, it has no work done on it by, nor 
does it do any work against, its surroundings consequent 
upon its contraction or expansion. Hence, using partial 
differentials and representing the condition of constraint 
by a subseript symbol, 

1! (88)p = (8U) n+ (80) 
By the second law, the system will be in equilibrium when 
it cannot do work upon its surroundings, that is, when 
its volume is maintained constant; for if any spontaneous 
change occurs in it, it must be such that if carried out 
reversibly it would do work. Hence (6W), cannot be 
positive: it can only be zero or negative. Therefore 
(SU) p»— 788), which is equal to — (dW), must be either 
zero or positive. Putting 

wv = U—TS, or 

(BW)r= (6U)p—1(85)r, 
it follows that the equilibrium condition is (dy)p 2 0, 
which can be the case only when yf is a minimum. 

To , defined by the equation above, the name of ‘ free 
energy’ is given; whence it follows that an tsothermal 
esometric system is in equilibrium when its free energy is a 
minimum. | 

II. Lsothermal isobaric systems: thermodynamic potential a 
minimum, Isothermal isobaric systems are of the greatest 
practical importance. Temperature is maintained constant, 
by means of a thermostat and the pressure is throughout 
that, say, of the atmosphere. The thermostat, as before, 
may be assumed of a kind which can give out or absorb 


Some Thermodynamical Considerations 183 


heat without doing work: a pressure regulator, thermally 
insulated from its surroundings, keeps the pressure constant 
at a value py. The simple system will now be in equilibrium 
when the extended system, comprising the simple one, the 
‘thermostat and the regulator, cannot do work upon the 
surroundings, that is, when the system does no work in 
excess of that done on the pressure regulator. The quantity 
@W in the original equation, 
: TdS = dU +dVW, 
must now be written in two parts, p (dv) 7, done on the 
regulator, and (6), ,, done on the surroundings, or 
T' (88), Pp ee (OU )-, » +p (dv) 7, Dp ste (OW )p. p° 
Then for equilibrium, 
(OW) 7, » = 9 or is negative ; 
therefore 
(OU) 7,» +p (8) 7, »p—T(88)z, » 2 9. 
Writing ¢ = U+pv—TS8 
and hence (0¢)p, 5 = (0U Ja, » +2 (80) 7, »— T'(38)p, » 
this condition becomes 
(8¢)2, » 2 0. 

To ¢, defined by the equation above, the name ‘ thermo- 
dynamic potential’ is given; whence it follows that az 
asothermal-isobarice system is in equilibrium when its thermo- 
dynamic potential is a minimum. 

By similar methods the corresponding conditions for 
equilibrium in isentropic and other systems may be 
deduced, but these are unimportant compared with the 
cases quoted. 

The remarks already made respecting the impossibility 
of calculating entropy in the polyphase systems met with 
in practice apply both to free energy and thermodynamic 
potential. Their applications can therefore be only quali- 
tative, but in the case of thermodynamic potential such 


184, Some Thermodynamical Considerations 


applications have proved of much value as general guides, 
since the equilibrium of heterogeneous systems is most 
readily amenable to experimental treatment when the 
systems are maintained at constant temperature and pres- , 
sure. Heterogeneous equilibria of this type have therefore 
been the most fully investigated of all, as has been shown 
often enough in the preceding: pages. 

Hence some further consideration of this function is 
desirable in connexion with the specific types of system 
with the equilibria of which the phase rule deals, namely, 
homogeneous phases, such as solids, liquids and mixtures 
of gases, the state of which is determined only when 
pressure, temperature and composition are known. | 

Thermodynamic potential and phase composition. When, 
as is usual, temperature and pressure are maintained con- 
stant, interest centres round variations in composition ; 
and it is desirable to seek an expression, in terms of the 
¢-function and the concentrations of individual components, 
for the conditions determining such spontaneous variations. 
The procedure to be followed will be the same as that 
adopted in the deduction of the ¢-function. A chosen 
system, with the necessary thermostats and pressure regu- 
lators, will be considered to be isolated and the condition 
of equilibrium taken to be in accordance with the second 
law, that in which any deviation, if conducted reversibly, 
would require work to be done on the system by its 
surroundings. 

As a simple and sufficient case, a system may be con- 
sidered containing two homogeneous solutions 4 and B of 
different compositions, maintained by suitable arrange- 
ments at common temperature and pressure. The masses 
of the several components in A are 4,, 4,, Az,..., 4,, and 
the corresponding components in B, b,, B,, B,,..., By. 


Some Thermodynamical Considerations 185 


For these two solutions to attain equilibrium it may be 
necessary for one to receive from the other definite amounts 
of the various components. For simplicity it will be 
assumed that only one component passes from B to A, that 
is, that 4, and #6, vary while 4,, 4,,..., 4, and B,, B,, 
..., B, remain constant. For the transfer of a small 
quantity 66, there will be a change in the sum of the 
energies U/, ea U, of the original systems, of the entropies 
S, and S, and the volumes V, and Vp. 

This change must be in accordance with the fundamental 
law 6Q=6U + OW, or putting ~ = 68, and rearranging, 
6U = T8S — dV. 

W may be divided into two parts, namely, work done 
against the pressure regulators included in the system, and 
work done against surroundings. ‘The complete expression 
for the energy change in this case is therefore given by 

O(U, + Up)e = 18(S4 + Spoi— po(Va + Valo — (OW eo, 
where subscript C indicates constancy of 7, p, A, ..., 4, and 
B,,..., B,. Rearranging terms, 

— (8W)o = (Uy — TS, + 2Vado + 8(Uy — TS, + pVe)o 

= (d¢4)o + (8€z)e: 

where ¢4 is the thermodynamic potential of the varied 
component in solution 4 and ¢, of that in B. 

These functions will be dependent, respectively, upon 
T, p and the masses of components, so that 


epee ap+ 264 .dp+ 4 oC Wik ae. elie, 





owe “Op | 6A, 6A, 2 
and 
Hine oe an + So a, ap+ 52 ay ieee seh, 
n 


But the assumption has been nee that the only variables 
are A, and B,. Hence all the differentials vanish except 


186 Some Thermodynanical Considerations 


dA, and dB,, where dA, is the amount of component (1) 
passing from 6 to A, written above as éd,. 
dB, is equal and of opposite sign, — 64,. Hence 


and : dép= _ ep gy. 


ba: S708 
6 6 
or, putting HA, = oe and = 4g, = oe 
1 
we have — (OW) = (pa, — Hp,) 0Ay. 


If now the passage of some B, to 4,, that is, the increase 
of A, by 64,, occur spontaneously, dW, by the second law, 
must be positive. 

Hence py, — fg, is negative since 64, is a 
therefore He, 7 HA: 

Chemical Potential, To the quantity pu the name of 
‘chemical potential’ is given and the result just obtained 
may be put in the following words : 

For a component to pass spontaneously from one system 
to another the thermodynamic condition is that its chemical 
potential in the one be greater than that in the other. 
Hence, also, equilibrium in respect to any component is 
possible between two systems only when its respective 
chemical potentials in the two are identical, 


The chemical potential sf of a component 4, of any 


system may be defined as the change in thermodynamic 
potential which the system suffers when, to a very large 
amount of it, there is added unit mass (gramme, gramme- 
molecule or other unit arbitrarily selected) of the component, 


~Y 


Some Thermodynamical Considerations 187 


the pressure, temperature and masses of other components 
being kept constant, and the total amount of the system 
taken being so large that the added unit of mass is 
relatively negligible. 

It was this quantity to which reference was made in the 
foot-note to p. 17, Chapter I. What were termed ‘ phase- 
equilibrium equations’ in the deduction of the Phase Rule, 
were, fundamentally, expressions of the equalities of the 
chemical potentials possessed by any one component in the 
various phases present in equilibrium. 

The theoretical significance of chemical potential is 
obviously very great: its practical quantitative application 
is not yet possible. For a thorough account of hetero- 
geneous equilibria from the thermodynamical standpoint 
the student is referred to the original work of Willard 
Gibbs, Scientific Papers, i, pp. 62-100. 


CHAPTER VIII 


A DISCUSSION ON BINARY AND TERNARY 
SYSTEMS OF MIXED CRYSTALS WITH IL- 
LUSTRATIONS OF THE GRAPHICAL USE 
OF THE ¢FUNCTION. 


AttHoucHu thermodynamic potential, or as it is usually 
termed, the ¢-function, cannot be calculated quantitatively 
for such phases of matter as have been dealt with in earlier 
chapters, yet it is possible to make use of it qualitatively to 
reach certain general conclusions of considerable interest. 

An appropriate example, which it is proposed to discuss 
in some detail, is its application as a guide to the possible 
types of mixed-crystal formation in binary systems. Before 


188 Binary and Ternary Systems 


passing, however, to this particular subject, attention will 
be directed to the very great theoretical significance, and 
the wide generality, of mixed-crystal (or solid-solution) 
formation. 

Compositions of solid phases. Reference has already been 
made in the Preface to the fact that in a binary system, 
and also in higher systems, an alteration in composition of 
solid phase must be regarded as the invariable effect of an 
alteration in composition of liquid phase, if equilibrium 
between the two be maintained. This means, in effect, 
that for every series (or curve) of liquid solutions there 
corresponds an equally definite series (or curve) of solid 
solutions. In many cases the latter is so exceedingly 
short that experimental analysis fails to reveal it, and the 
curve is, for practical purposes, a point. 

In Chapters IV, V and VI the question of composition of 
single-phase solids has been deliberately ignored, though, 
in ignoring this, it has frequently been impossible to avoid 
language not strictly justifiable. Thus, one has spoken of 
the single composition of a point in relations where a 
statement of a succession of compositions could alone be 
justified. To put the argument for this in a form rather 
different from that adopted in the Preface, it is only 
necessary to re-examine a couple of cases which have been 
considered, respectively, in Chapter IV (Figure 26, p. 74) 
and Chapter V (Figure 67, p. 120). 

If it were strictly true that every solution along the 
curve ECF in Figure 26 could be in equilibrium with one 
and the same solid of composition C’, it would follow that 
at a temperature such as 7 the two solutions ¢ and / could 
coexist with one another. This, of course,is not so. In 
whatever proportions these two be brought together, a 
phase reaction occurs and a solid phase appears. 


of Mined Crystals 189 


So, too, in Figure 67, if the solutions @ and J on the 
curve ed could exist in equilibrium with precisely the same 
solid S, then they would be able to coexist with one 
another; and the same applies to all solutions along ed. 
They cannot do so. Again, if solution e were in equi- 
librium with the solids 4 and S, and solution d with the 
solids S and C, it would follow that a univariant condensed 
system of solution e, solution d, solid 4, solid S and solid C 
(five phases in all) would be able to exist. This is in very 
definite contradiction to the Phase Rule, so that it is 
impossible to avoid the conclusion that the composition 
of a solid phase varies, however little, with that of the 
liquid with which it is in equilibrium. 

The justification for the course that has been followed, 
in omitting any reference to slight changes in compositions 
of solids, lies in the fact that the student will probably 
find it easier to take that course and then amend (or add 
to) it in certain particulars than to take the strictly 
accurate course from the beginning. 

The importance of a consideration of mixed crystals will 
now be apparent, though it will hardly be necessary to 
discuss a great number of examples. It will simplify 
matters to regard the pressure always as constant and 
sufficient to prevent vapour formation in all systems about 
to be mentioned, since alterations in equilibria between 
solids and liquids, brought about by small variations of 
vapour pressure, will be quite negligible: in other words, 
such systems may very reasonably be treated as ‘con- 
densed ’. 

A deduction of all the types for binary systems was first 

made by Roozeboom in a classic paper.* His deduction 
was based on a qualitative use of thermodynamic potential, 
* Zeit. phys. Chem., 1899, 30, 385. 


190 Bunary and Ternary Systems 


and it is not possible to do better than to follow his 
method. nee 
Binary Systems. At any temperature the thermodynamic 
potential for each possible mixture of any two components 
will have a definite value if the mixture be solid and 
another definite value if it be liquid. ‘The equilibrium 
state will be that which possesses the minimum value of ¢. 
The two sets of values 
may at some given 
temperature be identi- 
cal for all mixtures for 
such similar compo- 
nents as optical iso- 
mers, and they may be 
considerably alike for 
two components of 
equal melting-point : 
but in general they 
will differ more or less 
from one another. As 
a first simple case, it 


a function —> 











A ———« Bswill be assumed that 


Figure 107. ‘ 
the curves connecting 


composition and ¢ value for solid and liquid mixtures, 
respectively, are similar in form but not identical: also 
that the melting-points of the components differ, that of 
B being the higher. For brevity these two curves will be 
denoted by ¢, for the solid, and ¢,, for the liquid, mixture. 

For all temperatures above the melting-point of B, the 
¢s curve will lie wholly above ¢,, as shown in Figure 107, 

The thermodynamic potential of every mixture is here- 
less in the liquid than in the solid state; that is to say, all 
mixtures are liquid or else metastable. With fall of tem- 


of Mixed Crystals 191 


- 


perature, both sets of ¢ values decrease, but ¢, more than 
'¢, At the melting-point of 56, where solid and liquid 
B are in equilibrium, the two curves start from the same 
point at 100 per cent. Bas shown diagrammatically in section 
I (i) of Figure 109. All mixtures containing any 4 at all 
are liquid. At lower temperatures the ¢y curve continues 
to fall more quickly than ¢;, and between the melting- 
points of B and 4 cuts 
it progressively lower 
and lower, as shown in 
sections I (11) and I 
(iii) of Figure 109. 

The phase relation- 
ships in such cases may 
be deduced from the 
general diagrammatic 
type represented in 
Figure 108. 

Itisat onceapparent 3 
from this figure that 2 
the states represented 
by a8Z and U'LZ are 
unstable relatively to 
those of a’ XZ and YZ, respectively. At 4 the thermo- 
dynamic potentials of solid and liquid mixtures of com- 
position z are identical: hence these must exist in 
equilibrium. Therefore at this temperature all mixtures 
from 4 to z are liquid and from £ to z are solid. The 
mixture z can exist as solid and as liquid, that is, this 
temperature is its melting-point. 

But although 7 represents an equilibrium state it is 
evident that the equilibrium is not stable. An infini- 
tesimally small displacement along the ¢ curve will not at 


af 






1 
a2 aS ee ee 

yk Die VS ays B 
Figure 108. 


192 Binary and Ternary Systems 


once be compensated by an equal and opposite displacement 


along the ¢, curve: for although a (where OC is a small 
change in composition) might happen to be equal to ase 


it has the opposite sign. Hence the change of thermo- 
dynamic potential brought about by the solidification of 
an infinitesimal amount of the liquid cannot be com- 
pensated by the melting of a corresponding amount of 
solid. On the contrary, any such solidification would at 
once upset the original unstable equilibrium, and’ the 
liquid and solid phases would pass in composition and 
thermodynamic potential to X and Y respectively. These 
points are those of contact of the common tangent to the 
¢y and ¢; curves, and not necessarily the turning-points of 
the curves. 
= at rome at Y; 

both are positive in the figure and the condition for stable — 
equilibrium is attained. 

Hence the possible systems from d to 6 at this tem- 
perature are the following :—From pure 4 to composition 
# all mixtures are liquid: from # to y there is a separation 
into phases, liquid of composition # and solid of composition 
y: from y to pure B all mixtures are solid. The propor- 
tions of the two phases separating between # and y are 
readily calculable : thus the mixture z would give zy/yx of 
liquid # and za/ay of solid y. 

At the melting-point of 4 the ¢ y curve les wholly 
below ¢,;, the two starting from the same point at 100 per 
cent. 4d. Tor this temperature, section I (iv) of Figure 109 
holds. Below it, pure components and all mixtures of the 
two can exist only as solids. 





of Mixed Crystals 193 


If now one plot for every temperature the compositions 
of the liquid and solid phases (such as # and y in Figure 108) 
in equilibrium at it, one obtains such a diagram as section I 
(vi) of Figure 109, a diagram the type of which may thus 
be deduced immediately from theoretical considerations 
of possible forms of the ¢-curves for solid and liquid 
mixtures. It shows a division according to temperature 
and composition into three fields. In the upper one all 
mixtures are homogeneous liquids. The middle one is 
bounded above by a curve for liquids (commonly termed the 
liquidus), below by one for solid phases (the solidus), Any 
mixture_of composition and temperature represented by a 
point in it passes into a heterogeneous equilibrium between 
solid and liquid phases. At any point in the area below’ 
the solidus curve the conditions are such that only a (single- 
phase) solid solution can exist. 

A comparison between this case and that-of Figures 21 
or 22 in Chapter IV will show the complete analogy 
which exists in two-component systems giving continuous 
series of equilibria between mixed crystals and liquids on 
the one hand and between liquids and vapours on the other. 

If the set of curves in column I, Figure 109, has been 
understood, those in columns II and III will not offer 
difficulty. 

In column II the ¢y curve is more sharply rounded than 
the ¢;, curve: in III the-opposite is assumed to be the case. 
Sections II (vi) and ITI (vi) give the deduced temperature- 
composition relations. Again the likeness to liquid-vapour 
systems of maximum and minimum boiling-points is clearly 
evident. The whole discussion of fractional distillation 
applied in the latter cases to the separation of the two 
components may be transferred, mutatis mutandis, to separa- 
tion by fractional crystallization in the former. 

2568 N 


Binary and Ternary Systens 


194 























FiGgurE 109. 


of Mixed Crystals 195 


In eases I, I] and III of Figure 109 the curves ¢, and 
¢z have been assumed to possess one turning-point only: 
but a case such as that represented in Figure 110 is clearly 
a possibility. 

Here are two turning-points, and a similar argument to 
that applied to Figure 108 will show that if \Y be the 
common tangent to the two sections, all mixtures between 
wand y, with ¢ values 
lying on XZY, can be 
inunstable equilibrium 
only, the one possible 
stable state being the 
equilibrium between 
two phases of com- 
positions # and y. The 
series of sections given 
in columns IV and V 
of Figure 109 show 
how one may deduce 
from the relative posi- 
tions of ¢, curves of 
this type and simple ~ ea y B 

Figure 110. 
¢; convex curves two 
typical cases where gaps appear in the series of mixed 
crystals, that is, where the series is not continuous from 
A to B as in the eases I, IT and III. 

In case IV the ¢, curve is taken to be such that as it comes 
from above to below ¢,, its two minima are on the JB side 
of the minimum of the latter. The liquidus curve in IV 
(vi) of Figure 109 then shows a change of curvature at £, 
which is a transition temperature. There is a gap at this 
temperature in the solidus curve, and the solution / is in 
equilibrium with two solid mixed crystals of compositions 

N2 


es function —+ 





196 Binary and Ternary Systems 


corresponding with / and G, respectively. Homogeneous 
single-phase mixed crystals cannot be formed from mix- 
tures lying between F and G: at equilibrium there is a 
separation into two solid 
phases. In the figure this 
gap is assumed to widen 
with fall of temperature, and 
the areas between which it 
lies are marked a and a’ to 
denote two different regions 
of miscibility. 

In case V the ¢y curve is 
assumed to be- such that as 
it crosses ¢; its minima lie 
on either side of the mini- 
mum of the latter. The 
liquidus curve, V (vi) in 
Figure 109, then shows 
a eutectic at # and again 
|| there is a gap in the misci- 
— bility of the solids. 

These assumptions regard- 
ing the forms of the ¢y and ¢;, 
curves have led to five types 
of composition-temperature 
diagram for two-component 
systems in which the solids 
may form mixed crystals. 
No types other than these are known, though IV and V 
may be obtained in different ways from those indicated, 
as for example when a melt may freeze to two kinds of 
mixed crystals. It will be a useful exercise for the student 
to follow the deductions of these curves from the ¢ values 











Fieure 111. 


of Mixed Crystals 197 


of the respective sets of solids and liquids in the way 
indicated in IV A and V A of Figure 111. 

It will also be a good exercise in interpretation of 
diagrams to follow the changes in such a mixture as J/ in 
section IV (vi) of Figure 109 as the temperature is raised 
to I’. For a fuller treatment of the whole subject the 
original memoir of Roozeboom (loc. cit.) should be studied. 

The possible cases of two gaps occurring in the series of 
mixed crystals should not be overlooked. 

Ternary Systems. A similar graphical treatment of the 
¢-function for other systems is possible, and for many 
ternary systems, both with and without mixed crystals, 
it has been worked out in a general way. A paper by 
van Rijn van Alkemade (Zed. phys. Chem., 1893, 10, 289) 
on the subject is well worth reading. Numerous applica- 
tions are discussed by Schreinemakers in Roozeboom’s Die 
heterogenen Gleichgewichte, vol. ili, part ii, pp. 284 et seq. 
Representing compositions in a triangular diagram, the 
values of the ¢-function may be plotted upwards at right 
angles and will lie on a surface which may be convex or 
concave downwards or changing from one to the other. 
From the form of the surface the course of the isotherm 
may be deduced. Since little more than an illustration of 
the principle is aimed at here, no further discussion will be 
given alone these lines; but it will be suitable just to 
follow this account of mixed crystals in binary systems 
with a short description of some of the many possible 
instances In ternary systems. 

Since in each of the three binary systems possible with 
three components 4, B and C, mixed crystals may, indeed 
must, be formed, there is, strictly speaking, only one case 
to consider. In practice, however, the variations from 
compositions of pure components may be of quite different 


198 Binary and Ternary Systems 


orders in the three cases: hence it is customary to distin- 
guish three general types, namely : 

I. One pair of components forms mixed crystals: the 
two others do not, or rather do so to an extent not per- 
ceptible experimentally. 

II. Two pairs, respectively, form mixed crystals: the 
third not. 

III. Each of the three pairs gives mixed crystals. 

A fourth type is possible where none of the three binary 
systems may form mixed crystals to more than an in- 
appreciable extent, but where these, if produced in recogniz- 
able amount, are always ternary in composition. 

For the present purpose it is quite sufficient to discuss 
one type only, and No. I, being the simplest, will best 
serve in illustration. 4 and C will be taken as the one 
pair of components which can form mixed crystals. From 
what has been said above, it follows at once that there may 
be five sub-types corresponding with the five possible types 
of system which 4 and C may form. The first of these, 
where 4 and C form a continuous series of mixed crystals 
the melting-points of which lie between those of 4 and C, 
will be discussed fully, and references will be made to the 
second and third, where this series shows a maximum or 
minimum as the case may be. 

A sketch of a solid model cannot very well be drawn to 
show clearly the variations of equilibrium from a tem- 
perature at which all mixtures are solid to one at which all 
are liquid, but the changes in these equilibria may be 
readily followed from Figure 112, where there are drawn 
isotherms at successive intervals of temperature. 

No.1 holds for temperatures below the melting-point of 
any stable solid, binary or ternary. All mixtures are there- 
fore solid and at equilibrium are conglomerates of solid B 


of Mixed Crystals 


and mixed crystals of 
A and C. A mixture 
a, for example, consists 
of ab/Lb parts of solid 
B side by side with 

—aB/Bbot mixedcrystals 
containing Cb/CA of A 
and Ad/CA of C. 

The lowest tempera- 
ture at which a liquid 
can exist will be, say, 
the eutectic tempera- 
ture of B and C, and 
immediately above it 
the isotherm will have 
the form of No. ii in 
Figure 112, and then 
in order iil, iv and v. 
Complexes falling 
within ced give only 
a single phase of un- 
saturatedliquid. Along 
cé are the liquids 
saturated with solid B: 
along de those satu- 
rated with mixed 
crystals the composi- 
tions of which vary 
continuously from C to 
fj. At e, therefore, is 
a liquid with which 
solid 6 and the mixed 
crystal / coexist, and 





oy) 
be 
= 
~~ 
3°) 
be 
7) 
os 
5 
rea 








Fieure 112. 


199 


200 Binary and Ternary Systems 


-any complex within the triangle Bef will separate into 
these three phases. This triangular area first increases in 
size and then diminishes, finally vanishing if and when ¢ 
falls on AB asin No. vi. At the same time, the triangle 
ABf, the remnant of the system No. i, vanishes. When ¢ 
lies on AB, the temperature is obviously that of the 
binary eutectic d—B. At this point the mixed crystals of 
A and C are in every case in equilibrium with respective 
liquids: in Nos. ii, iii, iv and v only one portion of the series 
was in equilibrium with liquids, the other being with 
solid C. 

Further rise in temperature causes an increase in the 
unsaturated liquid area ced which becomes cghd and steadily 
spreads over the whole triangle. At some temperature Syc 
will vanish, and at some other, above or below this, AAdC 
will also disappear after passing through the stages viii, 
ix and x. At the melting-point of C, assumed to be below 
that of 4, d and C will coincide giving the isotherm No. viii. 
This is the highest temperature at which every stage of 
solid mixed crystal from 4 to C is possible. Above it, 
as in 1x, some of the binary mixtures may be liquid only. 
Here id gives the compositions of liquids coexisting with 
successive mixed crystals from 4 to 4. Binary mixtures 
lying between 4 and d will not be solid, but will separate . 
into liquid of composition @ and the solid mixed erystal 4. 
From ¢@ to C, all mixtures are liquid. Ultimately 4, 4 and 
d all coincide at 4, and above this temperature no solid 
phase is possible with any proportions of the three com- 
ponents. 

If now, however, instead of the binary system 4-C 
giving a continuous series as in column I of Figure 109, it 
gives that of Figure 113, the ternary isotherms become 
somewhat modified. 


of Mixed Crystals 201 


Instead of e, in Figure 112, meeting 4B as in isotherm 
No. vi, between 4 and 8, it will reach it only at 4, and 






Temperature —» 





A Composition C 
Figure 113. 





B Tea Ee SAY 
Fieure 114. 
then at a temperature above this the isotherm will be of 
the type shown in Figure 114, 


202 


Binary and Ternary Systems 


The curve of liquids in equilibrium with mixed crystals 
is now Add, both ends being on the binary system 4-C, as 


Temperature —> 








Composition 


FieurE 115. 





2 
Figure 116. 


shown in Figure 
113. The series of 
the mixed crystals 
runs only from 4, 
to &, (see both 
Figures 113 and 
114). 4, 4, 4, d 
andall points on Add 
steadily approach 
one another, and at 
the temperature of 
m (Figure 118) the 
area A/d vanishes 
to the point Min 
Figure 114. 

Where, on the 
other hand, the 
binary system is 
that of Figure 115, 
it willeasily be seen 
that Figure 116 is 
the ternary iso- 
therm for a tem- 
perature below the 
melting-points of 
A and C but above 
the respective eu- 
tectics of A-B and 
B-C. 


Here mixed crystals can occur ‘only from C to #, and 
The corresponding liquids lie along /,d and 


from A to f,. 


of Mixed Crystals 203 


ih, respectively. With rise of temperature the area /,44 
diminishes and vanishes at 4, /,dC at C. 

One other point is worth noting. Returning to Figure 112, 
isotherms 11 to v, it will be seen that as e moves from BC 
to BA, it traces out what is really a ternary freezing-point 
curve. Such a curve may have a maximum or a minimum 
or neither, and in the subsequent stages of Figure 112 it 
was assumed to have 
neither. If, how- 
ever, 1t has a maxi- 
mum, then before 
the side 5d is 
reached there will 
be stages of the 
type represented in 
Figure 117. 

At the maximum 
point, ¢ and ¢’ will 
meet and the iso- 
therm thereafter will 
resolve itself into 
that of No. vii, Figure112. The various areas in Figure 117 
correspond with the following systems: 





Figure 117. 


ced and ced’ unsaturated liquids : 

Bee and Be'e’ ~~ solid B and solutions: 

Cfed and Af’e’d’ mixed crystals and solutions : 

Bff’ * mixed crystals and solid B: 

Bfe and Lf’e solid B, mixed crystal and solution. 


This is an interesting system in which to trace the 
changes in phase which follow changes in composition. 
One may at once read from the isotherm the successive 
equilibria which follow additions of C to a mixture of A 
and B of composition given, say, by the point w One may 


204: Binary and Ternary Systems 


also observe that addition in suitable proportions of un- 
saturated liquids in the areas ced and c’e’d’, respectively, 
may yield a mixture of mean composition lying in Bf’, 
which will be entirely solid, containing the two phases’B 
and mixed crystals of composition between / and 7”. 

When, on the other hand, the ternary freezing-point 
curve has a mini- 
mum, thenjust above 
this the isotherm has 
the form of Figure 
118, 

The area of unsatu- 
rated liquids 1s now 
that lying between ¢ 
and e’: along its one 
boundary the solid 
phase in equilibrium 
is 5, along the other, 

Fieure 118. mixed crystals from 
i tof’. The rest of the figure requires no explanation. : 

These ‘indications will serve to give an idea of the types 
of isotherm met in these ternary systems where mixed 
crystals may appear. It is obvious that a complete dis- 
cussion would be very lengthy, for the binary types IV 
and V of Figure 109 remain to be introduced into the 
ternary type I, while all five types, or combinations of 
them, are possible with ternary types ITI and III as well. 























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